DIMENSIONAL  ANALYSIS 


DIMENSIONAL  ANALYSIS 

BY 

P.  W.  BRIDGMAN 

II 

PEOFESSOE  OF  PHYSICS  IN  HAEVAED  UNIVEESITY 


NEW  HAVEN 

YALE^  UNIVERSITY  PRESS 

LONDON  :  HUMPHEEY  MILFOED   :  OXFOED  UNIVEESITY  PEESS 

MDCCCCXXII 


G° 


A 


COPYRIGHT,  1922,  BY 
YALE  UNIVERSITY  PRESS 


PEEFACE 

THE  substance  of  the  following  pages  was  given  as  a  series  of  five 
lectures  to  the  Graduate  Conference  in  Physics  of  Harvard  Univer- 
sity in  the  spring  of  1920. 

The  growing  use  of  the  methods  of  dimensional  analysis  in  tech- 
nical physics,  as  well  as  the  importance  of  the  method  in  theoretical 
investigations,  makes  it  desirable  that  every  physicist  should  have 
this  method  of  analysis  at  his  command.  There  is,  however,  nowhere 
a  systematic  exposition  of  the  principles  of  the  method.  Perhaps  the 
reason  for  this  lack  is  the  feeling  that  the  subject  is  so  simple  that 
any  formal  presentation  is  superfluous.  There  do,  nevertheless,  exist 
important  misconceptions  as  to  the  fundamental  character  of  the 
method  and  the  details  of  its  use.  These  misconceptions  are  so  wide- 
spread, and  have  so  profoundly  influenced  the  character  of  many 
speculations,  as  I  shall  try  to  show  by  many  illustrative  examples, 
that  I  have  thought  an  attempt  to  remove  the  misconceptions  well 
worth  the  effort. 

I  have,  therefore,  attempted  a  systematic  exposition  of  the  princi- 
ples underlying  the  method  of  dimensional  analysis,  and  have  illus- 
trated the  applications  with  many  examples  especially  chosen  to 
emphasize  the  points  concerning  which  there  is  the  most  common 
misunderstanding,  such  as  the  nature  of  a  dimensional  formula, 
the  proper  number  of  fundamental  units,  and  the  nature  of  dimen- 
sional constants.  In  addition  to  the  examples  in  the  text,  I  have 
included  at  the  end  a  number  of  practise  problems,  which  I  hope 
will  be  found  instructive. 

The  introductory  chapter  is  addressed  to  those  who  already  have 
some  acquaintance  with  the  general  method.  Probably  most  readers 
will  be  of  this  class.  I  have  tried  to  show  in  this  chapter  by  actual 
examples  what  are  the  most  important  questions  in  need  of  discus- 
sion. The  reader  to  whom  the  subject  is  entirely  new  may  omit  this 
chapter  without  trouble. 

I  am  under  especial  obligation  to  the  papers  of  Dr.  Edgar  Buck- 
ingham on  this  subject.  I  am  also  much  indebted  to  Mr.  M.  D.  Hersey 
of  the  Bureau  of  Standards,  who  a  number  of  years  ago  presented 
Dr.  Buckingham's  results  to  the  Conference  in  a  series  of  lectures. 

September,  1920. 


497291 


CONTENTS 


Chapter  I. 

Chapter  II. 

Chapter  III. 

Chapter  IV. 

Chapter  Y. 

Chapter  VI. 

Chapter  VII. 


Chapter  VIII. 

Problems 

Index 


Introductory      .... 

Dimensional  Formulas        .... 

On  the  Use  of  Dimensional  Formulas  in 
Changing  Units 

The  H  Theorem 

Dimensional  Constants  and  the  Number  of 
Fundamental  Units  .... 

Examples  Illustrative  of  Dimensional  Analy- 
sis ....... 

Applications  of  Dimensional  Analysis  to 
Model  Experiments.  Other  Engineering 
Applications  ..... 

Applications  to  Theoretical  Physics     . 


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CHAPTER  I 
INTRODUCTORY 

APPLICATIONS  of  the  methods  of  dimensional  analysis,  to  simple 
problems,  particularly  in  mechanics,  are  made  by  every  student  of 
physics.  Let  us  analyze  a  few  such  problems  in  order  to  refresh  our 
minds  and  get  before  us  some  of  the  questions  which  must  be 
answered  in  a  critical  examination  of  the  processes  and  assumptions 
underlying  the  correct  application  of  the  general  method. 

"We  consider  first  the  illustrative  problem  used  in  nearly  every 
introduction  to  this  subject,  that  of  the  simple  pendulum.  Our 
endeavor  is  to  find,  without  going  through  a  detailed  solution  of  the 
problem,  certain  relations  which  must  be  satisfied  by  the  various 
measurable  quantities  in  which  we  are  interested.  The  usual  proce- 
dure is  as  follows.  We  first  make  a  list  of  all  the  quantities  on  which 
the  answer  may  be  supposed  to  depend;  we  then  write  down  the 
dimensions  of  these  quantities,  and  then  we  demand  that  these  quan- 
tities be  combined  into  a  functional  relation  in  such  a  way  that  the 
relation  remains  true  no  matter  what  the  size  of  the  units  in  terms 
of  which  the  quantities  are  measured. 

Now  let  us  try  by  this  method  to  find  how  the  time  of  swing  of  the 
simple  pendulum  depends  on  the  variables  which  determine  the 
behavior.  The  time  of  swing  may  conceivably  depend  on  the  length 
of  the  pendulum,  on  its  mass,  on  the  acceleration  of  gravity,  and  on 
the  amplitude  of  swing.  Let  us  write  down  the  dimensions  of  these 
various  quantities,  using  for  our  fundamental  system  of  units  mass, 
length,  and  time.  In  the  dimensional  formulas  the  symbols  of  mass, 
length,  and  time  will  be  denoted  by  capital  letters,  raised  to  proper 
powers.  Our  list  of  quantities  is  as  follows : 

Name  of  Quantity.  Symbol.  Dimensional  Formula. 
Time  of  swing,  t  T 

Length  of  pendulum,  1  L 

Mass  of  pendulum,  m  M 

Acceleration  of  gravity,  g  LT~2 

Angular  amplitude  of  swing,         6  No  dimensions. 


„ ...DIMENSIONAL  ANALYSIS 

*^  »*"•*•:      •  A 

1  We  are' to1  find  t  as  a  function  of  1,  m,  g,  and  0,  such  that  the 

functional  relation  still  holds  when  the  size  of  the  fundamental 
units  is  changed  in  any  way  whatever.  Suppose  that  we  have  foupd 
this  relation  and  write 

Now  the  dimensional  formulas  show  how  the  various  fundamental 
units  determine  the  numerical  magnitude  of  the  variables.  The 
numerical  magnitude  of  the  time  of  swing  depends  only  on  the  size 
of  the  unit  of  time,  and  is  not  changed  when  the  units  of  mass  or 
length  are  changed.  Hence  if  the  equation  is  to  remain  true  when 
the  units  of  mass  and  length  are  changed  in  any  way  whatever,  the 
quantities  inside  the  functional  sign  on  the  right-hand  side  of  the 
equation  must  be  combined  in  such  a  way  that  together  they  are  also 
unchanged  when  the  units  of  mass  and  length  are  changed.  In  par- 
ticular, they  must  be  unchanged  when  the  size  of  the  unit  of  mass 
alone  is  changed.  Now  the  size  of  the  unit  of  mass  affects  only  the 
magnitude  of  the  quantity  m.  Hence  if  m  enters  the  argument  of 
the  function  at  all,  the  numerical  value  of  the  function  will  be 
changed  when  the  size  of  the  fundamental  unit  of  mass  is  changed, 
and  this  change  cannot  be  compensated  by  any  corresponding 
change  in  the  values  of  the  other  quantities,  for  these  are  not 
affected  by  changes  in  the  size  of  the  unit  of  mass.  Hence  the  mass 
cannot  enter  the  functional  relation  at  all.  This  shows  that  the 
relation  reduces  to 

Now  1  and  g  must  together  enter  the  function  in  such  a  way  that 
the  numerical  magnitude  of  the  argument  is  unchanged  when  the 
size  of  the  unit  of  length  is  changed  and  the  unit  of  time  is  kept 
constant.  That  is,  the  change  in  the  numerical  value  of  1  produced 
by  a  change  in  the  size  of  the  unit  of  length  must  be  exactly  com- 
pensated by  the  change  produced  in  g  by  the  same  change.  The 
dimensional  formula  shows  that  1  must  be  divided  by  g  for  this  to 
be  accomplished.  We  now  have 

Now  a  change  in  the  size  of  the  fundamental  units  produces  no 
change  in  the  numerical  magnitude  of  the  angular  amplitude,  be- 
cause it  is  dimensionless,  and  hence  0  may  enter  the  unknown  func- 
tion in  any  way.  But  it  is  evident  that  1/g  must  enter  the  function  in 


INTRODUCTORY  3 

such  a  way  that  the  combination  has  the  dimensions  of  T,  since 
these  are  the  dimensions  of  t  which  stands  alone  on  the  left-hand 
side  of  the  equation.  We  see  by  inspection  that  1/g  must  enter  as 
the  square  root  in  order  to  have  the  dimensions  of  T,  and  the  final 
result  is  to  be  written 


where  </>  is  subject  to  no  restriction  as  far  as  the  present  analysis 
can  go.  As  a  matter  of  fact,  we  know  from  elementary  mechanics, 
that  <£  is  very  nearly  a  constant  independent  of  0,  and  is  approxi- 
mately equal  to  2?r. 

A  question  may  arise  in  connection  with  the  dimensions  of  0.  We 
have  said  that  it  is  dimensionless,  and  that  its  numerical  magnitude 
does  not  change  when  the  size  of  the  fundamental  units  of  mass, 
length,  or  time  are  changed.  This  of  course  is  true,  but  it  does  not 
follow  that  therefore  the  numerical  magnitude  of  0  is  uniquely 
determined,  as  we  see  at  once  from  the  possibility  of  measuring  0  in 
degrees  or  in  radians.  Are  we  therefore  justified  in  treating  0  as  a 
constant  and  saying  that  it  may  enter  the  functional  relation  in  any 
way  whatever? 

Now  let  us  discuss  by  the  same  method  of  analysis  the  time  of 
small  oscillation  of  a  small  drop  of  liquid  under  its  own  surface 
tension.  The  drop  is  to  be  thought  of  as  entirely  outside  the  gravita- 
tional field,  and  the  oscillations  refer  to  periodic  changes  of  figure, 
as  from  spherical  to  ellipsoidal  and  back.  The  time  of  oscillation 
will  evidently  depend  on  the  surface  tension  of  the  liquid,  on  the 
density  of  the  liquid,  and  on  the  radius  of  the  undisturbed  sphere. 
We  have,  as  before, 

Name  of  Quantity.  Symbol.  Dimensional  Formula. 
Time  of  oscillation,  t  T 

Surface  tension,  s  MT~2 

Density  of  liquid,  d  MLr3 

Radius  of  drop,  r  L 

We  are  to  find  f  such  that 

t  =  f  (s,  d,  r) 

where  f  is  such  that  this  relation  holds  true  numerically  whatever 
the  size  of  the  fundamental  units  in  terms  of  which  t,  s,  d,  and  r  are 


4  DIMENSIONAL  ANALYSIS 

measured.  The  method  is  exactly  the  same  as  for  the  pendulum 
problem.  It  is  obvious  that  M  must  cancel  from  the  right-hand  side 
of  the  equation.  This  can  occur  only  if  s  and  d  enter  through  their 
quotient.  Hence 

t  =  f  (s/d,r). 

Now  since  L  does  not  enter  t,  L  cannot  enter  f .  Hence  s/d  and  r 
must  be  combined  in  such  a  way  that  L  cancels.  Since  L  enters  s/d 
to  the  third  power,  it  is  obvious  that  s/d  must  be  divided  by  the  cube 
of  r  in  order  to  get  rid  of  L.  Hence 

t  — f  (s/dr3). 

Now  the  dimensions  of  s/dr3  are  T~2.  The  function  must  be  of 
such  a  form  that  these  dimensions  are  converted  into  T,  which  are 
the  dimensions  of  the  left-hand  side.  Hence  the  final  result  is 


t  =  Const  V  dr8/s. 

That  is,  the  time  of  oscillation  is  proportional  to  the  three  halves 
power  of  the  radius,  to  the  square  root  of  the  density,  and  inversely 
to  the  square  root  of  the  surface  tension.  This  result  is  checked  by 
experiment.  The  result  was  given  by  Lord  Rayleigh  as  problem  7 
in  his  paper  in  Nature,  95,  66,  1915. 

Now  let  us  stop  to  ask  what  we  meant  when  in  the  beginning  we 
said  that  the  time  of  oscillation  will  ''depend"  only  on  the  surface 
tension,  density,  and  radius.  Did  we  mean  that  the  results  are  inde- 
pendent of  the  atomic  structure  of  the  liquid,  for  example  ?  Every- 
one will  admit  that  surface  tension  is  due  to  the  forces  between  the 
atoms  in  the  surface  layer  of  the  liquid,  and  will  depend  in  a  way 
too  complicated  for  us  at  present  to  exactly  express  on  the  shape 
and  constitution  of  the  atoms,  and  on  the  nature  of  the  forces  be- 
tween them.  If  this  is  true,  why  should  not  all  the  elements  which 
determine  the  forces  between  the  atoms  also  enter  our  analysis,  for 
they  are  certainly  effective  in  determining  the  physical  behavior? 
"We  might  justify  our  procedure  by  some  such  answer  as  this.  ' '  Al- 
though it  is  true  that  the  behavior  is  determined  by  a  most  com- 
plicated system  of  atomic  forces,  it  will  be  found  that  these  forces 
affect  the  result  only  in  so  far  as  they  conspire  to  determine  one 
property,  the  surface  tension."  This  implies  that  if  we  were  to 
measure  the  time  of  oscillation  of  drops  of  different  liquids,  differing 


INTRODUCTORY  5 

as  much  as  we  pleased  in  atomic  properties,  we  would  find  that  all 
drops  of  the  same  radius,  density,  and  surface  tension,  executed 
their  oscillations  in  the  same  time.  We  add  that  the  truth  of  this 
reply  may  be  checked  by  an  appeal  to  experiment.  But  our  critic 
may  not  even  yet  be  satisfied.  He  may  ask  how  we  were  sure  before- 
hand that  among  the  various  properties  of  the  liquids  of  which  the 
drop  might  be  composed  the  surface  tension  was  the  only  property 
.affecting  the  time  of  oscillation.  It  may  seem  quite  conceivable  to 
him  that  the  time  of  oscillation  might  depend  on  the  viscosity  or 
compressibility,  and  if  we  are  compelled  to  appeal  to  experiment, 
of  what  value  is  our  dimensional  analysis?  To  which  we  would  be 
forced  to  reply  that  we  have  indeed  had  a  wider  experimental 
experience  than  our  critic,  and  that  there  are  conditions  under 
which  the  time  of  oscillation  does  depend  on  the  viscosity  or  com- 
pressibility in  addition  to  the  surface  tension,  but  that  it  will  be 
found  as  a  matter  of  experiment  that  if  the  radius  of  the  drop  is 
made  smaller  and  smaller  there  is  a  point  beyond  which  the  com- 
pressibility will  be  found  to  play  an  imperceptibly  small  part,  and 
in  the  same  way  if  the  viscosity  of  the  liquid  is  made  smaller  and 
smaller,  there  will  also  be  a  point  beyond  which  any  further  reduc- 
tion of  the  viscosity  will  not  perceptibly  affect  the  oscillation  time. 
And  we  add  that  it  is  to  such  conditions  as  these  that  our  analysis 
applies.  Instead  of  appealing  to  direct  experiment  to  justify  our 
assertions,  we  might,  since  our  critic  is  an  intelligent  critic,  appeal 
to  that  generalization  from  much  experiment  contained  in  the  equa- 
tions of  hydrodynamics,  and  show  by  a  detailed  application  of  the 
equations  to  the  present  problem  that  compressibility  and  viscosity 
may  be  neglected  beyond  certain  limiting  conditions. 

We  shall  thus  ultimately  be  able  to  satisfy  our  critic  of  the  cor- 
rectness of  our  procedure,  but  to  do  it  requires  a  considerable  back- 
ground of  physical  experience,  and  the  exercise  of  a  discreet  judg- 
ment. The  untutored  savage  in  the  bushes  would  probably  not  be 
able  to  apply  the  methods  of  dimensional  analysis  to  this  problem 
and  obtain  results  which  would  satisfy  us. 


Now  let  us  consider  a  third  problem.  Given  two  bodies  of  masses 
mx  and  m2  in  empty  space,  revolving  about  each  other  in  a  spherical 
orbit  under  their  mutual  gravitational  attraction.  We  wish  to  find 
how  the  time  of  revolution  depends  on  the  other  variables.  We  make 
a  list  of  the  various  quantities  as  before. 


6  DIMENSIONAL  ANALYSIS 

Name  of  Quantity.  Symbol.  Dimensional  Formula. 

Mass  of  first  body,  n^                                M 

Mass  of  second  body,  m2                               M 

Distance  of  separation,  r                                L 

Time  of  revolution,  t                                T 

These  are  evidently  all  the  quantities  physically  involved,  because 
whenever  we  compel  two  bodies  of  masses  1%  and  m2  to  describe  a 
circular  orbit  about  each  other  under  their  own  gravitational  attrac- 
tion in  empty  space  at  a  distance  of  separation  r,  we  find  that  the 
time  of  revolution  is  always  the  same,  no  matter  what  the  material 
of  which  the  bodies  are  composed,  or  their  past  history,  chemical, 
dynamical,  or  otherwise.  Now  let  us  search  for  the  functional  rela- 
tion, writing, 

t=f  (m^m^r). 

We  demand  that  this  shall  hold  irrespective  of  the  size  of  the  funda- 
mental units.  A  moment's  examination  confuses  us,  because  the 
left-hand  side  involves  only  the  element  of  time,  and  the  elements 
of  the  right-hand  side  do  not  involve  the  time  at  all.  Our  critic  at 
our  elbow  now  suggests, ' '  But  you  have  not  included  all  the  elements 
on  which  the  result  depends;  it  is  obvious  that  you  have  left  out 
the  gravitational  constant."  "But,"  say  we,  "how  can  this  be? 
The  gravitational  constant  can  look  out  for  itself.  Nature  attends 
to  that  for  us.  It  is  undeniable  that  two  bodies  of  the  masses  mx  and 
m2  when  placed  at  a  distance  r  apart  always  revolve  in  the  same 
time.  We  have  included  all  the  physical  quantities  which  can  be 
varied."  But  our  critic  insists,  and  to  oblige  him  we  try  the  effect 
of  including  the  gravitational  constant  among  the  variables.  We  call 
the  gravitational  constant  G;  it  obviously  has  the  dimensions 
M"1  L3  T~2,  since  it  is  defined  by  the  equation  of  the  force  between 

m1m2 
two  gravitating  bodies,  force  =  G .  A  "constant"  which  has 

dimensions  and  therefore  changes  in  numerical  magnitude  when 
the  size  of  the  fundamental  units  changes  is  called  a  ' '  dimensional ' ' 
constant.  We  now  have  to  find  a  function  such  that  the  following 
relation  is  satisfied : 

t  =  f  (mltm2,  G,r). 

Now  this  functional  equation  is  not  quite  so  easy  to  solve  by  inspec- 
tion as  the  two  previous  ones,  and  we  shall  have  to  use  a  little 


INTRODUCTORY  7 

algebra  on  it.  Let  us  suppose  that  the  function  is  expressed  in  the 
form  of  a  sum  of  products  of  the  arguments.  Then  we  know  that  if 
the  two  sides  of  the  equation  are  to  remain  equal  no  matter  how  the 
fundamental  units  are  changed  in  magnitude,  the  dimensions  of 
every  one  of  the  product  terms  on  the  right-hand  side  must  be  the 
same  as  those  of  the  left-hand  side,  that  is,  the  dimensions  must  be 
T.  Assume  that  a  typical  product  term  is  of  the  form 

m«  mf  GV  r*. 
This  must  have  the  dimensions  of  T.  That  is, 

M«M0  (M-1  L3T-3)vL5  =  T. 
"Writing  down  the  conditions  on  the  exponents  gives 

a  +  /3-y  =  0 

3y  +  8  =  0 

-2y  =  l 

Hence 


The  values  of  a  and  ft  are  not  uniquely  determined,  but  only  a 
relation  between  them  is  fixed.  This  is  as  we  should  expect,  because 
we  had  only  three  equations  of  condition,  and  four  unknown  quan- 
tities to  satisfy  them  with.  The  relation  between  a  and  ft  shows  that 

m±  and  m2  must  enter  in  the  form  mT*[  ??i  J  ,  where  there  is  no 

\mi/ 

restriction  on  the  value  of  x.  Hence  our  unknown  function  is  of  the 
form 

f  =  2  A, 


G^mM~ 

where  x  and  Ax  may  have  any  arbitrary  values.  We  may  rewrite  f, 
by  factoring,  in  the  form 

~\m 
But  now  2  AX  ( — - )   ,  if  there  is  no  restriction  on  x  or  Ax,  is 


( — - 1   ,  if 

\w 


8  DIMENSIONAL  ANALYSIS 

merely  an  arbitrary  function  of  m  /m  ,  which  we  write  as  <£  [ — ?  J. 

\mi/ 
Hence  our  final  result  is 

L    — —          *    ^    ,    C 


That  is,  the  square  of  the  periodic  time  is  proportional  to  the 
cube  of  the  distance  of  separation,  and  inversely  as  the  gravita- 
tional constant,  other  things  being  equal. 

"We  can,  by  other  argument,  find  what  the  nature  of  the  unknown 
function  </>  is  in  one  special  case.  Suppose  a  very  heavy  central  body, 
and  a  satellite  so  light  that  the  two  together  revolve  approximately 
about  the  center  of  the  heavy  body.  It  is  obvious  that  under  these 
conditions  the  time  of  revolution  is  independent  of  the  mass  of  the 
satellite,  for  if  its  mass  is  doubled,  the  attractive  force  is  also 
doubled,  and  double  the  force  acting  on  double  the  mass  leaves  the 
acceleration,  and  so  the  time  of  revolution,  unaltered.  Therefore 
under  these  special  conditions  the  unknown  function  reduces  to  a 
constant,  if  we  denote  by  n^  the  mass  of  the  satellite,  and  the  rela- 
tion becomes 

t  =  Const      r      . 
G*m< 

This  relation  we  know  is  verified  by  the  facts  of  astronomy. 

Our  critic  seems,  therefore,  to  have  been  justified  by  the  results, 
and  we  should  have  included  the  gravitational  constant  in  the 
original  list.  We  are  nevertheless  left  with  an  uncomfortable  feeling 
because  we  do  not  see  quite  what  was  the  matter  with  our  argument, 
and  we  are  disturbed  by  the  foreboding  that  at  some  time  in  the 
future  there  may  perhaps  be  a  dimensional  constant  which  we  are 
not  clever  enough  to  think  of,  and  which  may  not  proclaim  the 
impossibility  of  neglecting  it  in  quite  such  uncompromising  tones 
as  the  gravitational  constant  in  the  example.  "We  are  afraid  that  in 
such  a  case  we  will  get  the  incorrect  answer,  and  not  know  it  until 
a  Quebec  bridge  falls  down. 

Beside  the  matter  of  dimensional  constants  the  last  problem  brings 
up  another  question.  "Why  is  it  that  we  had  to  assume  that  the  un- 
known function  could  be  represented  as  a  sum  of  products  of  powers 
of  the  independent  variables?  Certainly  there  'are  functions  in 
mathematics  which  cannot  be  represented  in  this  way.  Is  nature  to 


INTRODUCTORY  9 

be  arbitrarily  restricted  to  a  small  part  of  the  functions  which  one 
of  her  own  creatures  is  able  to  conceive  ? 

Consider  now  a  fourth  problem,  treated  by  Lord  Rayleigh  in 
Nature,  vol.  15,  66,  1915.  This  is  a  rather  famous  problem  in  heat 
transfer,  treated  before  Rayleigh  by  Boussinesq.  A  solid  body,  of 
definite  geometrical  shape,  but  variable  absolute  dimensions,  is 
fixed  in  a  stream  of  liquid,  and  maintained  at  a  definite  tempera- 
ture higher  than  the  temperature  of  the  liquid  at  points  remote  from 
the  body.  It  is  required  to  find  the  rate  at  which  heat  is  transferred 
from  the  body  to  the  liquid.  As  before,  we  make  a  list  of  the  various 
quantities  involved,  and  their  dimensions. 

Name  of  Quantity.  Symbol.           Dimensional  Formula. 

Rate  of  heat  transfer,  h  HT-1 

Linear  dimension  of  body,  a  L 

Velocity  of  stream,  v  LT"1 

Temperature  difference,  0  0 
Heat  capacity  of  liquid  per 

unit  volume,  c  HL~3  Or1 

Thermal  conductivity  of  liquid,    k  HL-1  T-1  0-1 

This  is  the  first  heat  problem  which  we  have  met,  and  we  have 
introduced  two  new  fundamental  units,  a  quantity  of  heat  (H), 
and  a  unit  of  temperature  (0).  It  is  to  be  noticed  that  the  unit  of 
mass  does  not  enter  into  the  dimensional  formulas  of  any  of  the 
quantities  in  this  problem.  If  we  desired,  we  might  have  introduced 
it,  dispensing  with  H  in  so  doing.  Now,  just  as  in  the  last  example, 
we  suppose  that  the  rate  of  heat  transfer,  which  is  the  quantity  in 
which  we  are  interested,  is  expressed  as  a  sum  of  products  of  powers 
of  the  arguments,  and  we  write  one  of  the  typical  terms 

Const  aa  00  w  cs  k*. 

As  before,  we  write  down  the  conditions  on  the  exponents  imposed 
by  the  requirement  that  the  dimensions  of  this  product  are  the  same 
as  those  of  h.  We  shall  thus  obtain  four  equations,  because  there 
are  four  fundamental  kinds  of  unit.  The  equations  are 

8  +  c  =      1  condition  on  exponent  of  H 
p  —  S  —  e  =      0  condition  on  exponent  of  0 
a_|_y_3S_€—     o  condition  on  exponent  of  L 
—  y  —  c  =  — 1  condition  on  exponent  of  T 

We  have  five  unknown  quantities  and  only  four  equations,  so  one 


10  DIMENSIONAL  ANALYSIS 

of  the  unknowns  must  remain  arbitrary.  Choose  this  one  to  be  y,  and 
solve  the  equations  in  terms  of  y.  This  gives 


•  =1-7 

Hence  the  product  term  above  becomes 
Const  a  0k 


\  k  / 

The  complete  solution  is  the  sum  of  terms  of  this  type.  As  before, 
there  is  no  restriction  on  the  constant  or  on  y,  so  that  all  these  terms 
together  coalesce  into  a  single  arbitrary  function,  giving  the  result 


Hence  the  rate  of  heat  transfer  is  proportional  to  the  temperature 
difference,  but  depends  on  the  other  quantities  in  a  way  not  com- 
pletely specifiable.  Although  the  form  of  the  function  F  is  not 
known,  nevertheless  the  form  of  the  argument  of  the  function  con- 
tains very  valuable  information.  For  instance,  we  are  informed 
that  the  effect  of  changing  the  velocity  of  the  fluid  is  exactly  the 
same  as  that  of  changing  its  heat  capacity.  If  we  double  the  velocity, 
keeping  the  other  variables  fixed,  we  affect  the  rate  of  heat  transfer 
precisely  as  we  would  if  we  doubled  the  heat  capacity  of  the  liquid, 
keeping  the  other  variables  constant. 

This  problem,  again,  is  capable  of  raising  many  questions.  One 
of  these  questions  has  been  raised  by  D.  Riabouchinsky  in  Nature, 
95,  591,  1915.  We  quote  as  follows: 

In   Nature   of  March   18   Lord   Rayleigh   gives   this   formula, 

(Q  rt  T^X 
),  considering  heat,  temperature,  length,  and  time 

as  four  "  independent ' '  quantities. 

If  we  suppose  only  three  of  these  quantities  are  "really  inde- 
pendent, ' '  we  obtain  a  different  result.  For  example,  if  the  tempera- 
ture is  defined  as  the  mean  kinetic  energy  of  the  molecules,  the 
principle  of  similitude*  allows  us  only  to  affirm  that 

h  =  ka0F( -,  c 

yji  *a 

*  Bayleigh  and  other  English  authors  use  this  name  for  dimensional  analysis. 


INTRODUCTORY  \  11 

That  is,  instead  of  obtaining  a  result  with  an  unknown  function 
of  only  one  argument,  we  should  have  obtained  a  function  of  two 
arguments.  Now  a  function  of  two  arguments  is  of  course  very  much 
less  restricted  in  its  character  than  a  function  of  only  one  argument. 
For  instance,  if  the  function  is  of  the  form  suggested  in  two  argu- 
ments, it  would  not  follow  at  all  that  the  effect  of  changing  the 
velocity  is  the  same  as  that  of  changing  the  heat  capacity.  Ria- 
bouchinsky,  therefore,  makes  a  real  point. 

Lord  Rayleigh  replies  to  Riabouchinsky  as  follows  on  page  644 
of  the  same  volume  of  Nature. 

The  question  raised  by  Dr.  Riabouchinsky  belongs  rather  to  the 
logic  than  the  use  of  the  principle  of  similitude,  with  which  I  was 
mainly  concerned.  It  would  be  well  worthy  of  further  discussion. 
The  conclusion  that  I  gave  follows  on  the  basis  of  the  usual  Fourier 
equations  for  the  conduction  of  heat,  in  which  temperature  and  heat 
are  regarded  as  sui  generis.  It  would  indeed  be  a  paradox  if  the 
further  knowledge  of  the  nature  of  heat  afforded  us  by  molecular 
theory  put  us  in  a  worse  position  than  before  in  dealing  with  a 
particular  problem.  The  solution  would  seem  to  be  that  the  Fourier 
equations  embody  something  as  to  the  nature  of  heat  and  tempera- 
ture which  is  ignored  in  the  alternative  argument  of  Dr.  Ria- 
bouchinsky. 

This  reply  of  Lord  Rayleigh  is,  I  think,  likely  to  leave  us  cold. 
Of  course  we  do  not  question  the  ability  of  Lord  Rayleigh  to  obtain 
the  correct  result  by  the  use  of  dimensional  analysis,  but  must  we 
have  the  experience  and  physical  intuition  of  Lord  Rayleigh  to 
obtain  the  correct  result  also  ?  Might  not  perhaps  a  little  examina- 
tion of  the  logic  of  the  method  of  dimensional  analysis  enable  us  to 
tell  whether  temperature  and  heat  are  * '  really ' '  independent  units 
or  not,  and  what  the  proper  way  of  choosing  our  fundamental 
units  is  ? 

Beside  the  prime  question  of  the  proper  number  of  units  to  choose 
in  writing  our  dimensional  formulas,  this  problem  of  heat  transfer 
raises  many  others  also  of  a  more  physical  nature.  For  instance, 
why  are  we  justified  in  neglecting  the  density,  or  the  viscosity,  or 
the  compressibility,  or  the  thermal  expansion  of  the  liquid,  or  the 
absolute  temperature?  We  will  probably  find  ourselves  able  to 
justify  the  neglect  of  all  these  quantities,  but  the  justification  will 
involve  real  argument  and  a  considerable  physical  experience  with 
physical  systems  of  the  kind  which  we  have  been  considering.  The 


12  DIMENSIONAL  ANALYSIS 

problem  cannot  be  solved  by  the  philosopher  in  his  armchair,  but 
the  knowledge  involved  was  gathered  only  by  someone  at  some  time 
soiling  his  hands  with  direct  contact. 

Finally,  we  consider  a  fifth  problem,  of  somewhat  different  char- 
acter. Let  us  find  how  the  electromagnetic  mass  of  a  charge  of  elec- 
tricity uniformly  distributed  throughout  a  sphere  depends  on  the 
radius  of  the  sphere  and  the  amount  of  the  charge.  The  charge  is 
considered  to  be  in  empty  space,  so  that  the  amount  of  the  charge 
and  the  radius  of  the  sphere  are  the  only  variables.  We  apply  the 
method  already  used.  The  dimensions  of  the  charge  (expressed  in 
electrostatic  units)  we  get  from  the  definition,  which  states  that  the 
numbers  measuring  the  magnitude  of  two  charges  shall  be  such  that 
the  force  between  them  is  equal  to  their  product  divided  by  the 
square  of  the  distance  between  them.  "We  accordingly  have  the 
following  table. 

Name  of  Quantity.  Symbol.  Dimensional  Formula. 

Charge,  e  M*  L*  T'1 

Radius  of  sphere,  r  L 

Electromagnetic  mass,  m  M 

We  now  write 

m  =  f  (e,r) 

and  try  to  find  the  form  of  f  so  that  this  relation  is  independent  of 
the  size  of  the  fundamental  units.  It  is  obvious  that  T  cannot  enter 
the  right-hand  side  of  the  equation,  since  it  does  not  enter  the  left, 
and  since  T  enters  th£  right-hand  side  only  through  e,  e  cannot 
enter.  But  if  e  does  not  enter  the  right-hand  side  of  the  equation,  M 
cannot  enter  either,  because  M  enters  only  into  e.  Hence  we  are 
left  with  a  contradiction  in  requirements  which -shows  that  the 
problem  is  impossible  of  solution.  But  here  again  our  Mephistophe- 
lean critic  suggests  that  we  have  left  out  a  dimensional  constant. 
We  demur ;  our  system  is  in  empty  space,  and  how  can  empty  space 
require  dimensional  constants?  But  our  critic  insists  that  empty 
space  does  have  properties,  and  when  we  push  him,  suggests  that 
light  is  propagated  with  a  definite  and  characteristic  velocity.  So 
we  try  again,  including  the  velocity  of  light,  c,  of  dimensions  LT~X, 
and  now  we  have 

m  =  f  (e,r,c). 


INTRODUCTORY  13 

We  now  no  longer  encounter  the  previous  difficulty,  but  imme- 
diately, with  the  help  of  our  experience  with  more  complicated 
examples,  find  the  solution  to  be 

m  =  Const  — . 
re8 

This  formula  may  be  verified  from  any  book  on  electrodynamics, 
and  our  critic  is  again  justified.  We  worry  over  the  matter  of  the 
dimensional  constant,  and  ultimately  take  some  comfort  on  recol- 
lecting that  c  is  also  the  ratio  of  the  electrostatic  to  the  electromag- 
netic units,  but  still  it  is  not  very  clear  to  us  why  this  ratio  should 
enter. 

On  reflecting  on  the  solutions  of  the  problems  above,  we  are 
troubled  by  yet  another  question.  Why  is  it  that  an  equation  which 
correctly  describes  a  relation  between  various  measurable  physical 
quantities  must  in  its  form  be  independent  of  the  size  of  the  funda- 
mental units?  There  does  not  seem  to  be  any  necessity  for  this  in 
the  nature  of  the  measuring  process  itself.  An  equation  is  a  descrip- 
tion of  a  phenomenon,  or  class  of  phenomena.  It  is  a  statement  in 
compact  form  that  if  we  operate  with  a  physical  phenomenon  in 
certain  prescribed  ways  so  as  to  obtain  a  set  of  numbers  describing 
the  results  of  the  operations,  these  numbers  will  satisfy  a  certain 
equation  when  substituted  into  it.  For  instance,  let  us  suppose  our- 
selves in  the  position  of  Galileo,  trying  to  determine  the  law  of  fall- 
ing bodies.  The  material  of  our  observation  is  all  the  freely  falling 
bodies  available  at  the  surface  of  the  earth.  We  use  as  our  instru- 
ments of  measurement  a  certain  unit  of  length,  let  us  say  the  yard, 
and  a  certain  unit  of  time,  let  us  say  the  minute.  With  these  instru- 
ments we  operate  on  all  falling  bodies  according  to  definite  rules. 
That  is,  we  obtain  all  the  pairs  of  numbers  we  can  by  associating 
for  any  and  all  of  the  bodies  the  distance  which  it  has  fallen  from 
rest  with  the  interval  of  time  which  has  elapsed  since  it  started  to 
fall.  And  we  make  a  great  discovery  in  the  observation  that  the 
number  expressing  the  distance  of  fall  of  any  body,  no  matter  what 
its  size  or  physical  properties  or  the  distance  it  has  fallen,  is  always 
a  fixed  constant  factor  times  the  square  of  the  number  expressing 
the  corresponding  elapsed  time.  The  numbers  which  we  have  ob- 
tained by  measurement  to  fit  into  this  relationship  were  obtained 
with  certain  definite  sized  units,  and  our  description  is  a  valid 


14  DIMENSIONAL  ANALYSIS 

description,  and  our  discovery  is  an  important  discovery  even  under 
the  restriction  that  distance  and  time  are  to  be  measured  with  the 
same  particular  units  as  those  which  we  originally  employed. 
We  can  write  our  discovery  in  the  form  of  an  equation 

s  =  Const  t2. 

Now  an  inhabitant  of  some  other  country,  who  uses  some  other 
system  of  units  equally  as  unscientific  as  the  yard  and  the  minute, 
hears  of  our  discovery,  and  tries  our  experiments  with  his  measur- 
ing instruments.  He  verifies  our  result,  except  that  he  must  use  a 
different  factor  of  proportionality  in  the  equation.  That  is,  the 
constant  depends  on  the  size  of  the  units  used  in  the  measurements, 
or  in  other  words,  is  a  dimensional  constant. 

The  verification  of  our  discovery  by  an  inhabitant  of  another 
country  is  reported  to  us,  and  we  retire  to  contemplate.  We  at 
length  offer  the  comment  that  this  is  as  it  should  be,  and  that  it 
could  not  well  be  otherwise.  We  offer  to  predict  in  advance  just 
how  the  constant  should  be  changed  to  fit  with  any  system  of  meas- 
urement, and  on  being  asked  for  details,  make  the  sophisticated 
suggestion  that  we  so  change  the  constant  as  to  exactly  neutralize 
any  change  in  the  numbers  representing  the  length  or  the  time,  so 
that  we  will  still  have  essentially  the  same  equation  as  before.  In 
particular,  if  the  unit  of  length  is  made  half  as  large  as  originally, 
so  that  the  number  measuring  a  certain  distance  of  fall  is  now 
twice  as  large  as  it  was  formerly,  we  multiply  the  constant  by  2  so 
as  to  compensate  for  the  factor  2  by  which  otherwise  the  left-hand 
side  of  the  equation  would  be  too  large.  Similarly  if  the  unit  of  time 
is  made  three  times  as  long  as  formerly,  so  that  the  number  express- 
ing the  duration  of  a  certain  free  fall  becomes  only  %  of  its 
original  value,  then  we  will  multiply  the  constant  by  9  to  com- 
pensate for  the  factor  %,  by  which  otherwise  the  right-hand  side  of 
the  equation  would  be  too  small.  In  other  words,  we  give  to  the  con- 
stant the  dimensions  of  plus  one  in  length,  and  minus  two  in  time, 
and  so  obtain  a  formula  valid  no  matter  what  the  size  of  the 
fundamental  units. 

This  experience  emboldens  us,  and  we  try  its  success  with  other 
much  more  complicated  systems.  For  instance,  we  make  observations 
of  the  height  of  the  tides  at  our  nearest  port,  using  a  foot  rule  to 
measure  the  height  of  the  water, 'and  a  clock  graduated  in  hours  as 


INTRODUCTORY  15 

the  time-measuring  instrument.  As  the  result  of  many  observations, 
we  find  that  the  height  of  water  may  be  represented  by  the  formula 

h  =  5  sin  0.5066  t. 

We  now  write  this  in  a  form  to  which  any  other  observer  using 
any  other  system  of  units  may  also  fit  his  measurements  by  intro- 
ducing two  dimensional  constants  into  the  formula,  which  takes 
the  form 

h  G!  =  5  sin  0.5066  C2 1, 

where  Cx  has  the  dimensions  of  If"1,  and  C2  has  the  dimensions 
of  T-1. 

This  result  immediately  suggests  a  generalization.  Any  equation 
whatever,  no  matter  what  its  form,  which  correctly  reproduces  the 
results  of  measurements  made  with  any  particular  system  of  units 
on  any  physical  system,  may  be  thrown  into  such  a  form  that  it  will 
be  valid  for  measurements  made  with  units  of  different  sizes,  by  the 
simple  device  of  introducing  as  a  factor  with  each  observed  quantity 
a  dimensional  constant  of  dimensions  the  reciprocal  of  those  of  the 
factor  beside  which  it  stands,  and  of  such  a  numerical  value  that  in 
the  original  system  of  units  it  has  the  value  unity. 

Of  course  it  may  often  happen  that  the  form  of  the  equation  is 
such  that  two  or  more  of  these  dimensional  constants  coalesce  into  a 
single  factor.  The  first  example  above  of  the  falling  body  is  one  of 
this  kind.  The  general  rule  just  given  would  have  led  to  the  intro- 
duction of  two  dimensional  constants,  one  with  s  on  the  left-hand 
side  of  the  equation,  and  the  other  with  t2  on  the  right-hand  side 
of  the  equation,  but  by  multiplying  up,  these  two  may  be  combined 
into  a  single  constant. 

Our  query  is  therefore  answered,  and  we  see  that  every  equation 
can  be  put  in  such  a  form  that  it  holds  no  matter  what  the  size  of 
the  fundamental  units,  but  we  are  left  in  a  greater  quandary  than 
ever  with  regard  to  dimensional  constants.  May  there  not  be  new 
dimensional  constants  appropriate  to  every  new  kind  of  problem, 
and  how  can  we  tell  beforehand  what  the  dimensional  constants  will 
be  ?  If  we  cannot  tell  beforehand  what  dimensional  constants  enter 
a  problem,  how  can  we  hope  to  apply  dimensional  analysis?  The 
dimensional  situation  thus  appears  even  more  hopeless  than  at  first, 
for  we  could  see  a  sort  of  reason  why  the  gravitational  constant 
should  enter  the  problem  of  two  revolving  bodies,  and  could  even 


16  DIMENSIONAL  ANALYSIS 

catch  a  glimmer  of  reasonableness  in  the  entrance  of  the  velocity 
of  light  into  the  problem  of  electromagnetic  mass,  but  it  is  cer- 
tainly difficult  to  discover  reasonableness  or  predictableness  if 
dimensional  constants  can  be  used  indiscriminately  as  factors  by  the 
side  of  every  measured  quantity. 

In  our  consideration  of  the  problems  above  we  have  also  made  one 
more  observation  that  calls  for  comment.  We  have  noticed  that  every 
dimensional  formula  of  every  measurable  quantity  has  always 
involved  the  fundamental  units  as  products  of  powers.  Is  this  neces- 
sary, or  may  there  be  other  kinds  of  dimensional  formulas  for  quan- 
tities measured  in  other  ways,  and  if  so,  how  will  our  methods  apply 
to  such  quantities? 

To  sum  up,  we  have  met  in  this  introductory  chapter  a  number 
of  important  questions  which  we  must  answer  before  we  can  hope 
to  use  the  methods  of  dimensional  analysis  with  any  certainty  that 
our  results  are  correct.  These  questions  are  as  follows. 

First  and  foremost,  when  do  dimensional  constants  enter,  and 
what  is  their  form? 

Is  it  necessary  that  the  dimensional  formula  of  every  measured 
quantity  be  the  product  of  powers  of  the  fundamental  kinds  of  unit  ? 

What  is  the  meaning  of  quantities  with  no  dimensions? 

Must  the  functions  descriptive  of  phenomena  be  restricted  to  the 
sum  of  products  of  powers  of  the  variables? 

What  kinds  of  quantity  should  we  choose  as  the  fundamentals  in 
terms  of  which  to  measure  the  others?  In  particular,  how  many 
kiixds  of  fundamental  units  are  there  ?  Is  it  legitimate  to  reduce  the 
number  of  fundamental  units  as  far  as  possible  by  the  introduction 
of  definitions  in  accord  with  experimental  facts? 

Finally,  what  is  the  criterion  for  neglecting  a  certain  kind  of 
quantity  in  any  problem,  as  for  example  the  viscosity  in  the  heat 
flow  problem,  and  what  is  the  character  of  the  result  which  we  will 
get,  approximate  or  exact?  And  if  approximate,  how  good  is  the 
approximation  ? 


CHAPTER  II 
DIMENSIONAL  FORMULAS 

IN  the  introductory  chapter  we  considered  some  special  problems 
which  raised  a  number  of  questions  that  must  be  answered  before 
we  can  hope  to  really  master  the  method  of  dimensional  analysis. 
Let  us  now  begin  the  formal  development  of  the  subject,  keeping 
these  questions  in  mind  to  be  answered  as  we  proceed. 

The  purpose  of  dimensional  analysis  is  to  give  certain  informa- 
tion about  the  relations  which  hold  between  the  measurable  quanti- 
ties associated  with  various  phenomena.  The  advantage  of  the 
method  is  that  it  is  rapid;  it  enables  us  to  dispense  with  making 
a  complete  analysis  of  the  situation  such  as  would  be  involved  in 
writing  down  the  equations  of  motion  of  a  mechanical  system,  for 
example,  but  on  the  other  hand  it  does  not  give  as  complete  infor- 
mation as  might  be  obtained  by  carrying  through  a  detailed  analysis. 

Let  us  in  the  first  place  consider  the  nature  of  the  relations  be- 
tween the  measurable  quantities  in  which  we  are  interested.  In  deal- 
ing with  any  phenomenon  or  group  of  phenomena  our  method  is 
somewhat  as  follows.  We  first  measure  certain  quantities  which  we 
have  some  reason  to  expect  are  of  importance  in  describing  ^e  \/ 
phenomenon.  These  quantities  which  we  measure  are  of  different 
kinds,  and  for  each  different  kind  of  quantity  we  have  a  different 
rule  of  operation  by  which  we  measure  it,  that  is,  associate  the 
quantity  with  a  number.  Having  obtained  a  sufficient  array  of 
numbers  by  which  the  different  quantities  are  measured,  we  search 
for  relations  between  these  numbers,  and  if  we  are  skillful  and 
fortunate,  we  find  relations  which  can  be  expressed  in  mathematical 
form.  We  are  usually  interested  preeminently  in  one  of  the 
measured  quantities  and  try  to  find  it  in  terms  of  the  others.  Under 
such  conditions  we  would  search  for  a  relation  of  the  form 

*i  =  f  (x2,x3,x4,  etc.) 

where  x15  x2,  etc.,  stand  for  the  numbers  which  are  the  measures 
of  particular  kinds  of  physical  quantity.  Thus  xx  might  stand  for 


18  DIMENSIONAL  ANALYSIS 

the  number  which  is  the  measure  of  a  velocity,  x2  may  stand  for  the 
number  which  is  the  measure  of  a  viscosity,  etc.  By  a  sort  of  short- 
hand method  of  statement  we  may  abbreviate  thi&  long-winded 
description  into  saying  that  x±  is  a  velocity,  but  of  course  it  really 
is  not,  but  is  only  a  number  which  measures  velocity. 

Now  the  first  observation  which  we  make  with  regard  to  a  func- 
tional relation  like  the  above  is  that  the  arguments  fall  into  two 
groups,  depending  on  the  way  in  which  the  numbers  are  obtained 
physically.  The  first  group  of  quantities  we  call  primary  quantities. 
These  are  the  quantities  which,  according  to  the  particular  set  of 
rules  of  operation  by  which  we  assign  numbers  characteristic  of 
the  phenomenon,  are  regarded  as  fundamental  and  of  an  irreducible 
simplicity.  Thus  in  the  ordinary  systems  of  mechanics,  the  funda- 
mental quantities  are  taken  as  mass,  length,  and  time.  In  any 
functional  relation  such  as  the  above,  certain  arguments  of  the 
function  may  be  the  numbers  which  are  the  measure  of  certain 
lengths,  masses,  or  times.  Such  quantities  we  will  agree  to  call 
primary  quantities. 

In  the  measurement  of  primary  quantities,  certain  rules  of  opera- 
tion must  be  set  up,  establishing  the  physical  procedure  by  which  it 
is  possible  to  measure  a  length  in  terms  of  a  particular  length  which 
we  choose  as  the  unit  of  length,  or  a  time  in  terms  of  a  particular 
interval  of  time  selected  as  the  standard,  or  in  general,  it  is  charac- 
teristic of  primary  quantities  that  there  are  certain  rules  of  proce- 
dure by  which  it  is  possible  to  measure  any  primary  quantity 
directly  in  terms  of  units  of  its  own  kind.  Now  it  will  be  found  that 
we  always  make  a  tacit  requirement  in  selecting  the  rules  of  opera- 
tion by  which  primary  quantities  are  measured  in  terms  of  quanti- 
ties of  their  own  kind.  This  requirement  for  measurement  of  length, 
for  example,  is  that  if  a  new  unit  of  length  is  chosen  let  us  say  half 
the  length  of  the  original  unit,  then  the  rules  of  operation  must  be 
such  that  the  number  which  represents  the  measure  of  any  particu- 
lar concrete  length  in  terms  of  the  new  unit  shall  be  twice  as  large 
as  the  number  which  was  its  measure  in  terms  of  the  original  unit. 
Very  little  attention  seems  to  have  been  given  to  the  methodology  of 
systems  of  measurement,  and  I  do  not  know  whether  this  charac- 
teristic of  all  our  systems  of  measurement  has  been  formulated  or 
not,  but  it  is  evident  on  examination  of  any  system  of  measurement 
in  actual  use  that  it  has  these  properties.  The  possession  of  this 
property  involves  a  most  important  consequence,  which  is  that  the 


DIMENSIONAL  FORMULAS  19 

ratio  of  the  numbers  expressing  the  measures  of  any  two  concrete 
lengths,  for  example,  is  independent  of  the  size  of  the  unit  with 
which  they  are  measured.  This  consequence  is  of  course  at  once 
obvious,  for  if  we  change  the  size  of  the  fundamental  unit  by  any 
factor,  by  hypothesis  we  change  the  measure  of  every  length  by  the 
reciprocal  of  that  factor,  and  so  leave  unaltered  the  ratio  of  the 
measures  of  any  two  lengths.  This  means  that  the  ratio  of  the 
lengths  of  any  two  particular  objects  has  an  absolute  significance 
independent  of  the  size  of  the  units.  This  may  be  put  into  the  con- 
verse form,  as  is  evident  on  a  minute 's  reflection.  If  we  require  that 
our  system  of  measurement  of  primary  kinds  of  quantity  in  terms 
of  units  of  their  own  kind  be  such  that  the  ratio  of  the  measures  of 
any  two  concrete  examples  shall  be  independent  of  the  size  of  the 
unit,  then  the  measures  of  the  concrete  examples  must  change 
inversely  as  the  size  of  the  unit. , 

Besides  primary  quantities,  there  is  another  group  of  quantities 
which  we  may  call  secondary  quantities.  The  numerical  measures 
of  these  are  not  obtained  by  some  operation  which  compares  them 
directly  with  another  quantity  of  the  same  kind  which  is  accepted 
as  the  unit,  but  the  method  of  measurement  is  more  complicated  and 
roundabout.  Quantities  of  the  second  kind  are  measured  by  making 
measurements  of  certain  quantities  of  the  first  kind  associated  with 
the  quantity  under  consideration,  and  then  combining  the  measure- 
ments of  the  associated  primary  quantities  according  to  certain  rules 
which  give  a  number  that  is  defined  as  the  measure  of  the  secondary 
quantity  in  question.  For  example,  a  velocity  as  ordinarily  defined 
is  a  secondary  quantity.  We  obtain  its  measure  by  measuring  a 
length  and  the  time  occupied  in  traversing  this  length  (both  of 
these  being  primary  quantities),  and  dividing  the  number  measur- 
ing the  length  by  the  number  measuring  the  time  (or  dividing  the 
length  by  the  time  according  to  our  shorthand  method  of  state- 
ment). 

Now  there  is  a  certain  definite  restriction  on  the  rules  of  opera- 
tion which  we  are  at  liberty  to  set  up  in  defining  secondary  quan- 
tities. We  make  the  same  requirement  that  we  did  for  primary  quan- 
tities, namely,  that  the  ratio  of  the  numbers  measuring  any  two 
concrete  examples  of  a  secondary  quantity  shall  be  independent  of 
the  size  of  the  fundamental  units  used  in  making  the  required 
primary  measurements.  That  is,  to  say  that  one  substance  is  twice 
as  viscous  as  another,  for  example,  or  that  one  automobile  is  travel- 


20  DIMENSIONAL  ANALYSIS 

ling  three  times  as  rapidly  as  another,  has  absolute  significance, 
independent  of  the  size  of  the  fundamental  primary  units. 

This  requirement  is  not  necessary  in  order  to  make  measurement 
itself  possible.  Any  rules  of  operation  will  serve  as  the  basis  of  a 
system  of  measurement  by  which  numbers  may  be  assigned  to 
phenomena  in  such  a  way  that  the  particular  aspect  of  the  phenome- 
non on  which  we  are  concentrating  attention  is  uniquely  denned  by 
the  number  in  conjunction  with  the  rules  of  operation.  But  the 
requirement  that  the  ratio  be  constant,  or  we  may  say  the  require- 
ment of  the  absolute  significance  of  relative  magnitude,  is  essential 
to  all  the  systems  of  measurement  in  scientific  use.  In  particular, 
it  is  an  absolute  requirement  if  the  methods  of  dimensional  analysis 
are  to  be  applied  to  the  results  of  the  measurements.  Dimensional 
analysis  cannot  be  applied  to  systems  which  do  not  meet  this 
requirement,  and  accordingly  we  consider  here  only  such  systems. 

It  is  particularly  to  be  noticed  that  the  line  of  separation  between 
primary  and  secondary  quantities  is  not  a  hard  and  fast  one  im- 
posed by  natural  conditions,  but  is  to  a  large  extent  arbitrary,  and 
depends  on  the  particular  set  of  rules  of  operation  which  we  find 
convenient  to  adopt  in  defining  our  system  of  measurement.  For 
instance,  in  our  ordinary  system  of  mechanics,  force  is  a  secondary 
quantity,  and  its  measure  is  obtained  by  multiplying  a  number 
which  measures  a  mass  and  the  number  which  measures  an  accelera- 
tion (itself  a  secondary  quantity).  But  physically,  force  is  perfectly 
well  adapted  to  be  used  as  a  primary  quantity,  since  we  know  what 
we  mean  by  saying  that  one  force  is  twice  another,  and  the  physical 
processes  are  known  by  which  force  may  be  measured  in  terms  of 
units  of  its  own  kind.  It  is  the  same  way  with  velocities ;  it  is  pos- 
sible to  set  up  a  physical  procedure  by  which  velocities  may  be 
added  together  directly,  and  which  makes  it  possible  to  measure 
velocity  in  terms  of  units  of  its  own  kind,  and  so  to  regard  velocity 
as  a  primary  quantity.  It  is  perhaps  questionable  whether  all  kinds 
of  physical  quantity  are  adapted  to  be  treated,  if  it  should  suit  our 
convenience,  as  primary  quantities.  Thus  it  is  not  at  once  obvious 
whether  a  physical  procedure  could  be  set  up  by  which  two  viscosi- 
ties could  be  compared  directly  with  each  other  without  measuring 
other  kinds  of  quantity. 

But  this  question  is  not  essential  to  our  progress,  although  of 
great  interest  in  itself,  and  need  not  detain  us.  The  facts  are  simply 
these.  The  assigning  of  numerical  magnitudes  to  measurable  quanti- 


DIMENSIONAL  FORMULAS  21 

ties  involves  some  system  of  rules  of  operation  such  that  the  quanti- 
ties fall  into  two  groups,  which  we  call  primary  and  secondary. 

We  have  stated  that  the  requirement  of  the  absolute  significance 
of  relative  magnitude  imposes  definite  restrictions  on  the  operations 
by  which  secondary  quantities  may  be  measured  in  terms  of  primary 
quantities.  Let  us  formulate  this  restriction  analytically.  We  call 
the  primary  quantities  in  terms  of  which  the  secondary  quantity 
are  measured  a,  ft,  y,  etc.  Measurements  of  the  primary  quantities 
are  combined  in  a  certain  way  to  give  the  measure  of  the  secondary 
quantity.  We  represent  this  combination  by  the  functional  symbol  f, 

putting  the  secondary  quantity  =  f  (a,  /?,  y, ) .  Now  if  there 

are-  two  concrete  examples  of  the  secondary  quantity,  the  associated 
primary  quantities  have  different  numerical  magnitudes.  Let  us 
denote  the  set  associated  with  the  first  of  the  concrete  examples  by 
the  subscript  1,  and  that  with  the  second  set  by  the  subscript  2. 

Then  f  (c^, /?15  y1? )  will  be  the  measure  of  the  first  concrete 

example,  and  f  (a2,  /?2,  y2, )  will  be  the  measure  of  the  second. 

We  now  change  the  size  of  the  fundamental  units.  We  make  the 
unit  in  terms  of  which  a  is  measured  1/xth  as  large.  Then,  as  we 
have  shown,  the  number  measuring  a  will  be  x  times  as  large,  or  xa. 
In  the  same  way  make  the  unit  measuring  /?  1/yth  as  large,  and  the 
measuring  number  becomes  y/?.  Since  our  rule  of  operation  by  which 
the  numerical  measure  of  the  secondary  quantity  is  obtained  from 
the  associated  primary  quantities  is  independent  of  the  size  of  the 
primary  units,  the  number  measuring  the  secondary  quantity  now 
becomes  f  {xa,  y/?, ) .  The  measures  of  the  two  concrete  exam- 
ples of  the  secondary  quantity  will  now  be  f  (xa±,  y^, )  and 

f  (xa2,y/?2, ). 

Our  requirement  of  absolute  significance  of  relative  magnitude 
now  becomes  analytically 


This  relation  is  to  hold  for  all  values  of  a1?  ft, ,  a2,  ft, 

andx,  y,  z, . 

We  desire  to  solve  this  equation  for  the  unknown  function  f. 
Rewrite  in  the  form 

f  (**,,  yft, )  =  f  (xa,,  yft,  -  -  — )  X  *<a"&'  * 


22  DIMENSIONAL  ANALYSIS 

Differentiate  partially  with  respect  to  x.  Use  the  notation  fx  to 
denote  the  partial  derivative  of  the  function  with  respect  to  the  first 
argument,  etc.  Then  we  obtain 

%  f  i  (**!,  yft,  ----  )  =  oa  fi  (xo2,  y/?2,  -       -)  X  f   °"" 
Now  put  x,  y,  z,  etc.,  all  equal  to  1.  Then  we  have 


>  f   (/&,  ----  )  f   (a,,  ft,  ----  )* 

This  is  to  h61d  for  all  values  of  a^  /31?  ----  and  a2,  £2,  - 
Hence,  holding  a2,  /?2,  ----  fast,  and  allowing  ax,  j81?  ----  to  vary, 
we  have 

a  dt 

-  g-  ==  Const, 

or 

JL  df  _    Const 

T  ^  "  ~v~> 

which  integrates  to 


The  factor  C±  is  a  function  of  the  other  variables  /?,  y,  ----  . 

The  above  process  may  now  be  repeated,  differentiating  partially 
successively  with  respect  to  y,  z,  etc.,  and  integrating.  The  final 
result  will  obviously  be 

f  =C 


where  a,  b,  c,  -  -  —  and  C  are  constants. 

Every  secondary  quantity,  therefore,  which  satisfies  the  require- 
ment of  the  absolute  significance  of  relative  magnitude  must  be 
expressible  as  some  constant  multiplied  by  arbitrary  powers  of  the 
primary  quantities.  We  have  stated  that  it  is  only  secondary  quanti- 
ties of  this  kind  which  are  used  in  scientific  measurement,  and  no 
other  kind  will  be  considered  here. 

We  have  now  answered  one  of  the  questions  of  the  introductory 
chapter  as  to  why  it  was  that  in  the  dimensional  formulas  the 
fundamental  units  always  entered  as  products  of  powers. 

It  is  obvious  that  the  operations  by  which  a  secondary  quantity 
is  measured  in  terms  of  primary  quantities  are  defined  mathemati- 
cally by  the  coefficient  C,  and  by  the  exponents  of  the  powers  of 


DIMENSIONAL  FORMULAS  23 

the  various  primary  quantities.  For  the  sake  of  simplicity,  the 
coefficient  is  almost  always  chosen  to  be  unity,  although  there  is  no 
necessity  in  such  a  choice.  There  are  systems  in  use  in  which  the 
factor  is  not  always  chosen  as  unity.  Thus  the  so-called  rational  and 
the  ordinary  electrostatic  units  differ  by  a  factor  V^w-  Any  differ- 
ences in  the  numerical  coefficient  are  not  important,  and  are  always 
easy  to  deal  with,  but  the  exponents  of  the  powers  are  a  matter  of 
vital  importance.  The  exponent  of  the  power  of  any  particular 
primary  quantity  is  by  definition  the  ' '  dimension ' '  of  the  secondary 
quantity  in  that  particular  primary  quantity. 

The  "dimensional  formula"  of  a  secondary  quantity  is  the  aggre- 
gate of  the  exponents  of  the  various  primary  quantities  which  are 
involved  in  the  rules  of  operation  by  which  the  secondary  quantity 
is  measured.  In  order  to  avoid  confusion,  the  exponents  are  asso- 
ciated with  the  symbols  of  the  primary  quantities  to  which  they 
belong,  that  symbol  being  itself  written  as  raised  to  the  power  in 
question. 

For  example,  a  velocity  is  measured  by  definition  by  dividing  a 
certain  length  by  a  certain  time  (do  not  forget  that  this  really  means 
dividing  the  number  which  is  the  measure  of  a  certain  length  by 
the  number  which  measures  a  certain  time).  The  exponent  of  length 
is  therefore  plus  one,  and  the  exponent  of  time  is  minus  one,  and 
the  dimensional  formula  of  velocity  is  LT"1.  In  the  same  way  a  force 
is  defined  in  the  ordinary  Newtonian  mechanical  system  as  mass 
times  acceleration.  The  dimensions  of  force  are  therefore  equal  to 
mass  times  the  dimensions  of  acceleration.  The  dimensions  of  accel- 
eration are  obtained  from  its  definition  as  time  rate  of  change  of 
velocity  to  be  LT~2,  which  gives  for  the  dimensions  of  force  MLT~2. 

It  is  to  be  noticed  that  the  dimensions  of  any  primary  quantity 
are  by  a  simple  extension  of  the  definition  above  merely  the  dimen- 
sional symbol  of  the  corresponding  primary  quantity  itself. 

It  is  particularly  to  be  emphasized  that  the  dimensions  of  a  pri- 
mary quantity  as  defined  above  have  no  absolute  significance  what- 
ever, but  are  defined  merely  with  respect  to  that  aspect  of  the  rules 
of  operation  by  which  we  obtain  the  measuring  numbers  associated 
with  the  physical  phenomenon.  The  dimensional  formula  need  not 
even  suggest  certain  essential  aspects  of  the  rules  of  operation.  For 
example,  in  the  dimensional  formula  of  force  as  mass  times  accelera- 
tion, the  fact  is  not  suggested  that  force  and  acceleration  are  vectors, 
and  the  components  of  each  in  the  same  direction  must  be  com- 


24  DIMENSIONAL  ANALYSIS 

pared.  Furthermore,  in  our  measurements  of  nature,  the  rules  of 
operation  are  in  our  control  to  modify  as  we  see  fit,  and  we  would 
certainly  be  foolish  if  we  did  not  modify  them  to  our  advantage 
according  to  the  particular  kind  of  physical  system  or  problem  with 
which  we  are  dealing.  We  shall  in  the  following  find  many  problems 
in  which  there  is  an  advantage  in  choosing  our  system  of  measure- 
ment, that  is,  our  rules  of  operation,  in  a  particular  way  for  the 
particular  problem.  Different  systems  of  measurement  may  differ 
as  to  the  kinds  of  quantity  which  we  find  it  convenient  to  regard 
as  fundamental  and  in  terms  of  which  we  define  the  others,  or  they 
may  even  differ  in  the  number  of  quantities  which  we  choose  as 
fundamental.  All  will  depend  on  the  particular  problem,  and  it  is 
our  business  to  choose  the  system  in  the  way  best  adapted  to  the 
problem  in  hand. 

There  is  therefore  no  meaning  in  saying  "the"  dimensions  of  a 
physical  quantity,  until  we  have  also  specified  the  system  of  meas- 
urement with  respect  to  which  the  dimensions  are  determined.  This 
is  not  always  kept  clearly  in  mind  even  by  those  who  in  other  condi- 
tions recognize  the  relative  nature  of  a  dimensional  formula.  As  for 
example,  Buckingham  in  Phys.  Rev.  4,  357,  1914,  says:  "...  Mr. 
Tolman's  reasoning  is  based  on  the  assumption  that  absolute  tem- 
perature has  the  dimensions  of  energy,  and  this  assumption  is  not 
permissible."  Tolman,1  in  a  reply,  admitted  the  correctness  of  this 
position.  My  position  in  this  matter  would  be  that  Mr.  Tolman  has 
a  right  to  make  the  dimensions  of  temperature  the  dimensions  of 
energy  if  it  is  compatible  with  the  physical  facts  (as  it  seems  to  be) 
and  if  it  suits  his  convenience. 

This  view  of  the  nature  of  a  dimensional  formula  is  directly 
opposed  to  one  which  is  commonly  held,  and  frequently  expressed. 
It  is  by  many  considered  that  a  dimensional  formula  has  some 
esoteric  significance  connected  with  the  ' '  ultimate  nature "  of  an 
object,  and  that  we  are  in  some  way  getting  at  the  ultimate  nature 
of  things  in  writing  their  dimensional  formulas.  Such  a  point  of 
view  sees  something  absolute  in  a  dimensional  formula  and  attaches 
a  meaning  to  such  phrases  as  "really"  independent,  as  in  Ria- 
bouchinsky's  comments  on  Lord  Rayleigh's  analysis  of  a  certain 
problem  in  heat  transfer.  For  this  point  of  view  it  becomes  impor- 
tant to  find  the  "true"  dimensions,  and  when  the  "true"  dimen- 
sions are  found,  it  is  expected  that  something  new  will  be  suggested 
about  the  physical  properties  of  the  system.  To  this  view  it  is 


DIMENSIONAL  FORMULAS  25 

repugnant  that  there  should  be  two  dimensional  formulas  for  the 
same  physical  quantity.  Often  a  reconciliation  is  sought  by  the 
introduction  of  so-called  suppressed  dimensions.  Such  speculations 
have  been  particularly  fashionable  with  regard  to  the  nature  of 
the  ether,  but  so  far  as  I  know,  no  physical  discovery  has  ever  fol- 
lowed such  speculations;  we  should  not  expect  there  would  if  the 
view  above  is  correct. 

In  the  appendix  of  this  chapter  are  given  a  number  of  quotations 
characteristic  of  this  point  of  view,  or  others  allied  to  it. 


26  DIMENSIONAL  ANALYSIS 


APPENDIX  TO  CHAPTER  II 

QUOTATIONS  ILLUSTRATING  VAKIOUS  COMMON  POINTS  OF  VIEW 
WITH  REGARD  TO  THE  NATURE  OF  DIMENSIONAL  FORMULAS 

R.  C.  TOLMAN,  Phys.  Rev.  9  :  251,  1917. 

.  .  .  our  ideas  of  the  dimensions  of  a  quantity  as  a  shorthand  re- 
statement of  its  definition  and  hence  as  an  expression  of  its  essential 
physical  nature. 

A.  W.  RUCKER,  Phil.  Mag.  27  :  104,  1889. 

IN  the  calculation  of  the  dimensions  of  physical  quantities  we  not 

infrequently  arrive  at  indeterminate  equations  in  which  two  or  more 

unknowns  are  involved.  In  such  cases  an  assumption  has  to  be  made, 

and  in  general  that  selected  is  that  one  of  the  quantities  is  an  ab- 

stract number.  In  other  words,  the  dimensions  of  that  quantity  are 

suppressed. 

The  dimensions  of  dependent  units  which  are  afterwards  deduced 
from  this  assumption  are  evidently  artificial,  in  the  sense  that  they 
do  not  necessarily  indicate  their  true  relations  to  length,  mass,  and 
time.  They  may  serve  to  test  whether  the  two  sides  of  an  equation 
are  correct,  but  they  do  not  indicate  the  mechanical  nature  of  the 
derived  units  to  which  they  are  assigned.  On  this  account  they  are 
often  unintelligible. 

W.  W.  WILLIAMS,  Phil.  Mag.  34  :  234,  1892. 

That  these  systems  (i.e.,  the  electrostatic  and  the  electromagnetic) 
are  artificial  appears  when  we  consider  that  each  apparently  ex- 
presses the  absolute  dimensions  of  the  different  quantities,  that  is, 
their  dimensions  only  in  terms  of  L,  M,  and  T  ;  whereas  we  should 
expect  that  the  absolute  dimensions  of  a  physical  quantity  could  be 
expressed  in  only  one  way.  Thus  from  the  mechanical  force  between 
two  poles  we  get 


and  this,  being  a  qualitative  as  well  as  a  quantitative  relation, 
involves  the  dimensional  equality  of  the  two  sides.  ...  In  this  way 
we  get  two  different  absolute  dimensions  for  the  same  physical 
quantity,  each  of  which  involves  a  different  physical  interpreta- 
tion. .  .  . 

The  dimensional  formula  of  a  physical  quantity  expresses  the 
numerical  dependence  of  the  unit  of  that  quantity  upon  the  funda- 
mental and  secondary  units  from  which  it  is  derived,  and  the  indices 
of  the  various  units  in  the  formula  are  termed  the  dimensions  of  the 


DIMENSIONAL  FORMULAS  27 

quantity  with  respect  to  those  units.  When  used  in  this  very  re- 
stricted sense,  the  formulae  only  indicate  numerical  relations  be- 
tween the  various  units.  It  is  possible,  however,  to  regard  the  matter 
from  a  wider  point  of  view,  as  has  been  emphasized  by  Professor 
Riicker  in  the  paper  referred  to.  The  dimensional  formulae  may  be 
taken  as  representing  the  physical  identities  of  the  various  quanti- 
ties, as  indicating,  in  fact,  how  our  conceptions  of  their  physical 
nature  (in  terms,  of  course,  of  other  and  more  fundamental  con- 
ceptions) are  formed,  just  as  the  formula  of  a  chemical  substance 
indicates  its  composition  and  chemical  identity.  This  is  evidently 
a  more  comprehensive  and  fundamental  view  of  the  matter,  and 
from  this  point  of  view  the  primitive  numerical  signification  of  a 
dimensional  formula  as  merely  a  change  ratio  between  units  becomes 
a  dependent  and  secondary  consideration. 

The  question  then  arises,  what  is  the  test  of  the  identity  of  a 
physical  quantity  ?  Plainly  it  is  the  manner  in  which  the  unit  of  that 
quantity  is  built  up  (ultimately)  from  the  fundamental  units  L,  M, 
and  T,  and  not  merely  the  manner  in  which  its  magnitude  changes 
with  those  units. 

That  the  dimensional  formulae  are  regarded  from  this  higher 
standpoint,  that  is,  regarded  as  being  something  more  than  mere 
1 '  change  ratios ' '  between  units,  is  shown  by  the  fact  that  difficulties 
are  felt  when  the  dimensions  of  two  different  quantities,  e.g.,  couple 
and  work,  happen  to  be  the  same. 

S.  P.  THOMPSON,  Elementary  Lessons  in  Electricity  and  Magnetism, 
p.  352. 

It  seems  absurd  that  there  should  be  two  different  units  of  elec- 
tricity. 

R.  A.  FESSENDEN,  Phys.  Rev.  10 : 8,  1900. 

The  difference  between  the  dimensional  formula  and  the  qualita- 
tive formula  or  quality  of  a  thing  is  that,  according  to  the  defini- 
tions of  the  writers  quoted  above,  the  dimensions  "are  arbitrary, " 
are  "merely  a  matter  of  definition  and  depend  entirely  upon  the 
system  of  units  we  adopt, "  whilst  the  quality  is  an  expression  of 
the  absolute  nature,  and  never  varies,  no  matter  what  system  of 
units  we  adopt.  For  this  to  be  true,  no  qualities  must  be  suppressed. 

REFERENCES 

(1)  R.  C.  Tolman,  Phys.  Rev.  6,  1915,  p.  226,  footnote. 


CHAPTER  III 

ON  THE  USE  OF  DIMENSIONAL  FORMULAS  IN 
CHANGING  UNITS 

WE  saw  in  the  last  chapter  how  to  obtain  the  dimensional  formula 
of  any  quantity  in  terms  of  the  quantities  which  we  chose  by  defini- 
tion to  make  fundamental.  Our  method  of  analysis  showed  also  the 
connection  between  the  numerical  magnitude  of  the  derived  quantity 
and  the  fundamental  quantities.  Thus  if  length  enters  to  the  first 
power  in  the  dimensional  formula,  we  saw  that  the  number  measur- 
ing that  quantity  is  doubled  when  the  unit  of  length  is  halved,  or 
the  numerical  measures  are  inversely  as  the  size  of  the  unit,  raised 
to  the  power  indicated  in  the  dimensional  formula. 

Let  us  consider  a  concrete  example.  What  is  a  velocity  of  88  feet 
per  second  when  expressed  in  miles  per  hour  ?  The  dimensional  for- 
mula of  a  velocity  is  LT"1.  Now  if  our  unit  of  length  is  made  larger 
in  the  ratio  of  a  mile  to  a  foot,  that  is,  in  the  ratio  of  5280  to  1,  the 
velocity  will  be  multiplied  by  the  factor  1/5280,  because  length 
enters  in  the  dimensional  formula  to  the  first  power.  And  similarly, 
if  the  unit  of  time  is  made  larger  in  the  ratio  of  the  hour  to  the 
second,  that  is,  in  the  ratio  of  3600  to  1,  the  velocity  will  be  multi- 
plied by  the  factor  3600,  because  time  enters  the  dimensional  for- 
mula to  the  inverse  first  power.  To  change  from  feet  per  second  to 
miles  per  hour  we  therefore  multiply  by  3600/5280,  and  in  this 
particular  case  the  result  is  88  X  3600/5280,  or  60  miles  per  hour. 

Now  the  result  of  these  operations  may  be  much  contracted  and 
simplified  in  appearance  by  a  sort  of  shorthand.  We  write 

mile 


sec.  1  sec.  _J_  h 

3600 

88  3600  mile 
5280  hour 

=        60^?. 
hour 


FORMULAS  IN  CHANGING  UNITS  29 

A  little  reflection,  considering  the  relation  of  the  dimensional 
formula  to  the  operations  by  which  we  obtain  the  measuring  number 
of  any  physical  quantity,  will  at  once  show  that  this  procedure  is 
general,  and  that  we  may  obtain  any  new  magnitude  in  terms  of 
new  units  from  the  old  magnitude  by  using  the  dimensional  formula 
in  precisely  the  same  way.  This  method  of  use  of  the  dimensional 
formula  is  frequently  very  convenient,  and  is  the  simplest  and  most 
reliable  way  of  changing  units  with  which  I  am  acquainted. 

In  treating  the  dimensional  formula  in  this  way  we  have  endowed 
it  with  a  certain  substantiality,  substituting  for  the  dimensional 
symbol  of  the  fundamental  unit  the  name  of  the  concrete  unit 
employed,  and  then  replacing  this  concrete  unit  by  another  to  which 
it  is  physically  equivalent.  That  is,  we  have  treated  the  dimensional 
formula  as  if  it  expressed  operations  actually  performed  *on  physical 
entities,  as  if  we  took  a  certain  number  of  feet  and  divided  them  by 
a  certain  number  of  seconds.  Of  course,  we  actually  do  nothing  of 
the  sort.  It  is  meaningless  to  talk  of  dividing  a  length  by  a  time; 
what  we  actually  do  is  to  operate  with  numbers  which  are  the  meas- 
ure of  these  quantities.  We  may,  however,  use  this  shorthand  method 
of*  statement,  if  we  like,  with  great  advantage  in  treating  problems 
of  this  sort,  but  we  must  not  think  that  we  are  therefore  actually 
operating  with  the  physical  things  in  any  other  than  a  symbolical 
way.1 

This  property  of  the  dimensional  formula  of  giving  the  change  in 
the  numerical  magnitude  of  any  concrete  example  when  the  size  of 
the  fundamental  units  is  changed  makes  possible  a  certain  point 
of  view  with  regard  to  the  nature  of  a  dimensional  formula.  This 
view  has  perhaps  been  expressed  at  greatest  length  by  James  Thom- 
son in  B.  A.  Rep.  1878,  451.  His  point  of  view  agrees  with  that  taken 
above  in  recognizing  that  it  is  meaningless  to  say  literally  that  a 
velocity,  for  instance,  is  equal  to  a  length  divided  by  a  time.  We 
cannot  perform  algebraic  operations  on  physical  lengths,  just  the 
same  as  we  can  never  divide  anything  by  a  physical  time.  James 
Thomson  would  prefer,  instead  of  saying  velocit/  =  length/time, 
to  say  at  greater  length 

-     Change  ratio  of  velocity  =  Change  ratio  of  length 

Change  ratio  of  time 

Of  course  Thomson  would  not  insist  on  this  long  and  clumsy  expres- 
sion in  practise,  but  after  the  'matter  is  once  understood,  would 
allow  us  to  write  a  dimensional  formula  in  the  accustomed  way. 


30  DIMENSIONAL  ANALYSIS 

This  point  of  view  seems  perfectly  possible,  and  as  far  as  any 
results  go,  it  cannot  be  distinguished  from  that  which  I  have 
adopted.  However,  by  regarding  the  symbols  in  the  dimensional  for- 
mula as  reminders  of  the  rules  of  operation  which  we  used  physi- 
cally in  getting  the  numerical  measure  of  the  quantity,  it  seems  to 
me  that  we  are  retaining  a  little  closer  contact  with  the  actual 
physics  of  the  situation  than  when  we  regard  the  symbols  as  repre- 
senting the  factors  used  in  changing  from  one  set  of  units  to  another, 
which  after  all  is  a  more  or  less  sophisticated  thing  to  do,  and  which 
is  not  our  immediate  concern  when  first  viewing  a  phenomenon. 

Beside  the  sort  of  change  of  unit  considered  above,  in  which  we 
change  merely  the  sizes  of  the  fundamental  units,  there  is  another 
sort  of  change  of  unit  to  be  considered,  in  which  we  pass  from  one 
system  of  measurement  to  another  in  which  the  fundamental  units 
are  not  only  different  in  size,  but  different  in  character.2  Thus,  for 
example,  in  our  ordinary  system  of  units  of  Newtonian  mechanics 
we  regard  mass,  length,  and  time  as  the  fundamental  units,  but  it  is 
well  known  that  we  might  equally  regard  force,  length,  and  time 
as  fundamental.  We  may  therefore  expect  to  encounter  problems  of 
this  sort:  how  shall  we  express  a  kinetic  energy  of  10  gm  cm2  sec~2 
in  a  system  in  which  the  units  are  the  dyne,  the  cm,  and  the  sec  ? 

There  are  obviously  two  problems  involved  here.  One  is  to  find 
the  dimensional  formula  of  kinetic  energy  in  terms  of  force,  length, 
and  time,  and  the  other  is  to  find  the  new  value  of  the  numerical 
coefficient  in  that  particular  system  in  which  the  unit  of  force  is 
the  dyne,  the  unit  of  length  the  centimeter,  and  the  unit  of  time  the 
second. 

The  transformed  dimensional  formula  is  obtained  easily  if  we 
observe  the  steps  by  which  we  pass  from  one  system  to  the  other. 
The  transition  is  of  course  to  be  made  in  such  a  way  that  the  two 
systems  are  consistent  with  each  other.  Thus  if  force  is  equal  to  mass 
times  acceleration  in  one  system,  it  is  still  to  be  equal  to  mass  times 
'.  acceleration  in  the  other.  If  this  were  not  so,  we  would  be  concerned 
merely  with  a  formal  change,  and  the  thing  which  we  might  call 
force  in  the  one  system  would  not  correspond  to  the  same  physical 
complex  in  the  other.  This  relation  of  force  and  mass  in  the  two 
systems  is  maintained  by  an  application  of  simple  algebra.  In  the 
first  system  we  define  force  as  mass  times  acceleration,  and  in  the 
second  we  define  mass  as  force  divided  by  acceleration.  Thus  in 
each  system  the  secondary  quantity  is  expressed  in  terms  of  the 


FORMULAS  IN  CHANGING  UNITS  31 

fundamental  quantities  of  the  system,  and  the  two  systems  are 
consistent. 

The  correct  relation  between  the  dimensional  formulas  in  the  two 
systems  is  to  be  maintained  simply  by  writing  down  the  dimensional 
formulas  in  the  first  system,  and  then  inverting  these  formulas  by 
solving  for  the  quantities  which  are  to  be  regarded  as  secondary  in 
the  second  system.  In  the  special  case  considered,  we  would  have 
the  following  dimensional  formulas  : 

In  the  first  system  Force  =  MLT~2 

In  the  second  system  Mass  =  FL-1T2. 

The  transformation  of  the  numerical  coefficient  is  to  be  done 
exactly  as  in  the  example  which  we  have  already  considered  by 
treating  the  dimensional  symbols  as  the  names  of  concrete  things, 
and  replacing  the  one  to  be  eliminated  by  its  value  in  terms  of  the 
one  which  is  to  replace  it.  Thus  the  complete  work  associated  with 
the  problem  above  is  as  follows  : 

Igm   (lcm)*_ 

(i860)' 

We  have  to  find  the  transformation  equation  of  1  gm.  into  terms 
of  dynes,  cm,  and  sec.  Now 

1  dyne  =  1^lcm 
(1  sec)9 

Hence 

sec)a 


1  cm 
and  substituting 

1Q  Igm  (1cm)8  __  1Q  1  dyne  (1  sec)8  x  (1  cm)8 
(1  sec)8  1  cm  (1  sec)8 

=  10  dyne  cm, 

which  of  course  is  a  result  which  we  immediately  recognize  as  true. 

Let  us  consider  the  general  case  in  which  we  are  to  change  from 
a  system  of  units  in  which  the  fundamental  quantities  are  X1;  X2, 
and  X3  to  a  system  in  which  the  fundamentals  are  Y±,  Y2,  and  Y3. 

We  must  first  have  the  dimensional  formulas  of  Yt,  Y2,  and  Y3  in 
terms  of  X1?  X2,  and  X3.  Let  us  suppose  the  dimensions  of  Yt  are 
at,  a2,  and  a3,  those  of  Y2,  b1?  b2,  and  b3,  and  those  of  Y3,  ct,  c2,  and  c3 


32 


DIMENSIONAL  ANALYSIS 


in  Xlf  X2,  and  X3  respectively.  Then  in  any  concrete  case  we  may 
write 


where  the  C's  are  numerical  coefficients.  These  equations  are  to  be 
solved  for  the  X's  in  terms  of  the  Y's.  This  may  be  done  conven- 
iently by  taking  the  logarithm  of  the  equations,  giving, 

a±  log  Xt  +  a2  log  X2  +  a3  log  X3  =  log  C1  Yx 
bx  log  Xt  +  b2  log  X2  +  b3  log  X3  =  log  C2  Y2 
Cj.  log  Xx  +  c2  log  X2  +  c3  log  X3  =  log  C3  Y3 

These  are  algebraic  equations  in  the  logarithms,  and  may  be 
solved  immediately.  The  solution  for  X  is 


X  = 


_L_  A  c8c3^_  A  a2a3_^_   A 

'  '    y  (Ca  Ya)  a8aj  *    i  (C3  Y3)lb2bal  ' 


In  this  solution  A  stands  for  the  determinant  of  the  exponents. 


a2 
b 


The  values  of  X2  and  X3  are  to  be  obtained  from  the  value  for  Xx 
by  advancing  the  letters. 

Now  let  us  consider  an  example.  It  is  required  to  find  what  a 
momentum  of  15  tons  (mass)  miles/hour  becomes  in  that  system 
whose  fundamental  units  are  the  "2  Horsepower,"  the  "3  ft  per 
sec,"  and  the  "5  ergs."  This  ought  to  be  sufficiently  complicated. 
Introduce  the  abbreviations: 


Y2  for  the  "3  ft  per  sec" 
Y3  for  the  "5  ergs." 

1  Ib  (force)  1  ft 

Y±  =  2  H.P.  —  2  X  33000  -  ~  —  ~  — 

1mm 

In  the  first  place  we  have  to  change  Ibs  (force)  to  Ibs  (mass). 


FORMULAS  IN  CHANGING  UNITS  33 

Now  a  pound  force  is  that  force  which  imparts  to  a  mass  of  one 
pound  an  acceleration  of  32.17  ft/sec2. 
Hence 

1  Ib  (mass)  1  ft 

1  Ib  force  =  32.17 -r- 

(Isec)2 

and 

Ib  mass  1  ft       1ft 


(1  sec)3  1  min 


Y1==  66000  x   32.17 

(i  sec)' 

=  32.17  X  66000    ^TF  ton  (^ mi)3 
dnnnr  hour)3  -fa  hour 

=  2.962   x   104  i^^t 
hour8 

or,  writing  in  the  standard  form 

3.380  X  10~5  Y!  =  ton1  mi2  hour-3. 


Y  —  3—  —  3  ^A-7   mi    _  ^5    mi 
sec          -.rirW  hour       22  hour7 


Again 


or 

.4889  Y2  =  ton0  mi1  hour"1. 
And  again 

v  _  ,  _  ,.1  gin  (1cm)3  _     1.103  x  lO^tons  (6.214  x  10-6mi)2 

x    —  o  ergs  —  o —  D . 

(1  sec)'  („•„  hour)' 

or 

3.622  X  108  Y3  =  tons1  mi2  hour~2. 

We  rewrite  these  to  obtain  our  system  of  equations  in  the  stand- 
ard form 

3.380    x   10- 5  Y,  -  ton1  mi3  hour-3 

.4889  Y2  =  ton0  mi1  hour-1 

3.622     xlO8     Y3  =  ton1  mi2  hour-3. 

We  now  solve  for  the  ton,  mile,  and  hour  in  terms  of  Y1?  Y2,  and 
Y3.  We  first  find  the  determinant  of  the  exponents. 


A    = 


12-3 
01-1 
12—2 


__  -» 


This  is  pleasingly  simple. 


34  DIMENSIONAL  ANALYSIS 

The  general  scheme  of  solution  above  now  gives 

1  ton    =  (3.380  x  H)-6  Y,)0  (.4889  Ya)-3  (3.622  x  108  Y3)' 
1  mile  =  (  "  )->  (      "       )+»  (  "  )+1 

lhour  =  (  "  )•'(")'(  "  )+S 

or  simplifying, 

1  ton     =  15.16  x  108   Ya-2  Y3 
1  mile   =  5.223  x  10ia  Y^1  Y3  Ys 
1  hour  =  1.069  X  1013  Y,-1  Y3. 
And  finally 

tons  mi  _  15.16  x  5.223  x  10ao  Y,-a  Y,  Y,-1  Y,  Y3 

hour  1.069  x  1013  Y^'  Y3 

=  1.112  X  1010  Y,,-1  Y3 
5  ergs 

- L112  ><  10'°  ts/s 

which  is  the  answer  sought.  It  is  to  be  noticed  that  the  result  in- 
volves only  two  of  the  new  kind  of  unit  instead  of  three,  the  "2 
H.P. ' '  having  dropped  out.  This  of  course  will  not  in  general  be  the 
case.  It  might  at  first  sight  appear  that  we  might  take  advantage  of 
this  fact  and  eliminate  some  of  the  computation,  but  on  examination 
this  turns  out  not  to  be  the  case,  for  each  of  the  numerical  factors 
connecting  the  ton,  the  mile,  and  the  hour  with  the  new  units  is  seen 
to  be  involved  in  the  final  result. 

There  are  two  things  to  be  noticed  in  connection  with  the  above 
transformations.  In  the  first  place  it  is  not  always  possible  to  pass 
from  a  system  of  one  kind  of  units  to  a  system  of  another  kind,  but 
there  is  a  certain  relation  which  must  be  satisfied.  This  is  merely  the 
condition  that  the  equations  giving  one  set  of  units  in  terms  of  the 
other  shall  have  a  solution.  This  condition  is  the  condition  that  the 
transformed  equations,  after  the  logarithms  have  been  taken,  shall 
also  have  a  solution,  and  this  is  merely  the  condition  that  a  set  of 
algebraic  equations  have  a  solution.  This  condition  is  that  the  deter- 
minant of  the  coefficients  of  the  algebraic  equations  shall  not  vanish. 
Since  the  coefficients  of  the  algebraic  equations  in  the  logarithms  are 
the  exponents  of  the  original  dimensional  formulas,  the  condition  is 
that  the  determinant  of  the  exponents  of  the  dimensional  formulas 
for  one  system  of  units  in  terms  of  the  other  system  of  units  shall 
not  vanish. 

In  attempting  any  such  transformation  as  this,  the  first  thing  is 


FORMULAS  IN  CHANGING  UNITS  35 

to  find  whether  it  is  a  possible  transformation,  by  writing  down  the 
determinant  of  the  exponents.  If  this  vanishes,  the  transformation 
is  not  possible.  This  means  that  one  of  the  new  kinds  of  unit  in 
terms  of  which  it  is  desired  to  build  up  the  new  system  of  measure- 
ment is  not  independent  of  the  others.  Thus  in  the  example,  if 
instead  of  the  ' '  5  erg ' '  as  the  third  unit  of  the  new  system  we  had 
chosen  the  ' '  5  dyne, ' '  we  would  have  found  that  the  determinant  of 
the  exponents  vanishes,  and  the  transformation  would  not  have  been 
possible.  This  is  at  once  obvious  from  other  considerations.  For  a 
horse  power  is  a  rate  of  doing  work,  and  is  of  the  dimensions  of  the 
product  of  a  force  and  a  velocity,  and  the  second  unit  was  of  the 
dimensions  of  a  velocity,  so  that  the  proposed  third  unit,  which  was 
of  the  dimensions  of  a  force,  could  be  obtained  by  dividing  the  first 
unit  by  the  second,  and  would  therefore  not  be  independent  of  them. 
The  second  observation  is  that  the  new  system  of  units  to  which 
we  want  to  transform  our  measurement  must  be  one  in  which  there 
are  the  same  number  of  kinds  of  fundamental  unit  as  in  the  first 
system.  If  this  is  not  true,  we  shall  find  that,  except  in  special  cases, 
there  are  either  too  few  or  too  many  equations  to  allow  a  solution  for 
the  new  units  in  terms  of  the  old.  In  the  first  case  the  solution  is 
indeterminate,  and  in  the  second  no  solution  exists. 

REFERENCES 

(1)  D.  L.  Webster,  Sci.  46,  187,  1917. 
A.  Lodge,  Nat.  38,  281,  1888. 

(2)  A.  Buchholz,  Ann.  Phys.  51,  678,  1916. 


CHAPTER  IV 
THE  n  THEOREM 

IN  the  second  chapter  we  saw  that  the  dimensional  formulas  of  all 
the  quantities  with  which  we  shall  have  to  deal  are  expressible  as 
products  of  powers  of  the  fundamental  quantities.  Let  us  see  what 
inferences  this  enables  us  to  draw  about  the  forms  of  the  relations 
which  may  hold  between  the  various  measurable  quantities  con- 
nected with  a  natural  phenomenon. 

We  also  saw  in  the  second  chapter  that  at  least  sometimes  the 
functional  relation  will  involve  certain  so-called  dimensional  con- 
stants as  well  as  measurable  quantities.  We  met  two  examples  of 
dimensional  constants,  namely,  the  gravitational  constant,  and  the 
velocity  of  light  in  empty  space,  and  we  assigned  dimensional 
formulas  to  these  constants.  Now  it  is  most  important  to  notice  that 
these  two  dimensional  constants  had  dimensional  formulas  of  the 
type  proved  to  be  necessary  for  the  measurable  quantities,  namely, 
they  were  expressible  as  products  of  powers  of  the  fundamental 
quantities.  This  is  no  accident,  but  it  is  true  of  all  the  dimensional 
constants  with  which  we  shall  have  to  deal.  The  proof  can  best  be 
given  later  when  we  have  obtained  a  little  clearer  insight  into  the 
nature  of  a  dimensional  constant.  A  certain  apparent  exception,  the 
so-called  logarithmic  constant,  will  also  be  dealt  with  later.  We  may 
remark  here,  however,  that  one  class  of  dimensional  constant  must 
obviously  be  of  this  form.  We  saw  that  if  we  start  with  an  empirical 
equation  which  experimentally  has  been  found  to  be  true  from 
measurements  with  a  particular  set  of  units,  this  equation  can  be 
made  to  hold  for  all  sizes  of  the  units  by  the  device  of  introducing 
as  a  factor  with  each  measurable  quantity  a  dimensional  constant  of 
dimensions  the  reciprocal  of  those  of  the  measured  quantity.  Since 
the  dimensions  of  every  measured  quantity  are  products  of  powers, 
the  dimensions  of  the  reciprocal  must  also  be  products  of  powers, 
and  the  theorem  is  proved  for  this  restricted  class  of  dimensional 
constants.  We  will  for  the  present  accept  as  true  the  statement  that 
all  dimensional  constants  have  this  type  of  dimensional  formula. 


THE  H  THEOREM  37 

Now  let  us  suppose  that  we  have  a  functional  relation  between 
certain  measured  quantities  and  certain  dimensional  constants.  We 
shall  suppose  that  the  dimensional  formulas  of  all  these  quantities 
are  known,  including  the  dimensional  constants.  We  shall  further- 
more suppose  that  the  functional  relation  is  of  such  a  form  that  it 
remains  true  formally  without  any  change  in  the  form  of  the  func- 
tion when  the  size  of  the  fundamental  units  is  changed  in  any  way 
whatever.  An  equation  of  such  a  form  we  shall  call  a  "complete" 
equation.1  We  have  seen  that  it  is  by  no  means  necessary  that  an 
equation  should  be  a  complete  equation  in  order  to  be  a  correct  and 
adequate  expression  of  the  physical  facts,  although  the  contrary 
statement  is  almost  always  made,  and  is  frequently  made  the  basis 
of  the  proof  of  the  principle  of  dimensional  homogeneity  of  ' '  physi- 
cal ' '  equations.  Although  every  adequate  equation  is  not  necessarily 
complete,  we  have  seen  that  every  adequate  equation  can  be  made 
complete  in  a  very  simple  way,  so  that  the  assumption  of  complete- 
ness is  no  essential  restriction  in  our  treatment,  although  it  makes 
necessary  a  more  careful  examination  of  the  question  of  dimensional 
constants. 

The  assumption  of  the  completeness  of  the  equation  is  absolutely 
essential  to  the  treatment,  and  in  fact  dimensional  analysis  applies 
only  to  this  type  of  equation.  It  is  to  be  noticed  that  the  changes  of 
fundamental  unit  contemplated  in  the  complete  equation  are  re- 
stricted in  a  certain  sense.  We  may  change  only  the  size  of  the 
fundamental  units  and  not  their  character.  Thus,  for  example,  a 
complete  equation  which  holds  for  all  changes  in  the  size  of  the 
fundamental  units  as  long  as  these  units  are  units  of  mass,  length, 
and  time,  no  longer  is  true,  and  in  fact  becomes  meaningless  in 
another  system  of  units  in  which  mass,  force,  length,  and  time  are 
taken  as  fundamental. 

With  this  preliminary,  let  us  suppose  that  we  have  a  complete 
equation  in  a  certain  number  of  measurable  quantities  and  dimen- 
sional constants,  valid  for  a  certain  system  of  fundamental  units. 
Since  we  are  concerned  only  with  the  dimensional  formulas  of  the 
quantities  involved,  we  need  not  distinguish  in  our  treatment  the 
measurable  quantities  from  the  dimensional  constants.  We  will 

denote  the  variables  by  a,  /?,  y, to  n  quantities,  and  suppose 

there  is  a  functional  relation 


38  DIMENSIONAL  ANALYSIS 

The  expanded  meaning  of  this  expression  is  that  if  we  substitute 
into  the  functional  symbol  the  numbers  which  are  the  measures  of 
the  quantities  a,  ft  etc.,  the  functional  relation  will  be  satisfied.  We 
use  a  interchangeably  for  the  quantity  itself  and  for  its  numerical 
measure,  as  already  explained.  Now  the  fact  that  the  equation  is  a 
complete  equation  means  that  the  functional  relation  continues  to 
be  satisfied  when  we  substitute  into  it  the  numbers  which  are  the 
numerical  measure  of  the  quantities  a,  ft  ----  in  a  system  of  meas- 
urement whose  fundamental  units  differ  in  size  from  those  of  the 
fundamental  system  in  any  way  whatever.  Now  we  have  already 
employed  a  method,  making  use  of  the  dimensional  formulas,  for 
finding  how  the  number  measuring  a  particular  quantity  changes 
when  the  size  of  the  fundamental  units  changes.  This  was  the  sub- 
ject of  the  second  chapter.  Let  us  call  the  fundamental  units  m^  m2, 
m3,  etc.,  to  m  quantities,  and  denote  by  ax,  a2,  a3,  ----  etc.,  the 
dimensions  of  a,  by  ft,  ft>,  ft$,  ----  etc.,  the  corresponding  dimen- 
sions of  ft  etc.,  in  m1?  m2,  m3,  etc.,  respectively. 

We  now  decrease  the  size  of  the  fundamental  units  m^  m2,  etc., 
by  the  factors  xly  x2,  etc.  Then  the  numerical  measures  of  a,  ft  etc., 
in  terms  of  the  new  units,  which  we  will  call  a1,  /P,  etc.,  are,  as 
proved  in  Chapter  III,  given  by 


j8-=x{.xf.  ----  ft 


Now  since  the  equation  <f>  (a  ----  )  =0  is  a  complete  equation, 
it  must  still  hold  when  a1,  ft1,  ----  etc.,  are  substituted  for  a,  ft, 
----  .  That  is 


or 

*(X?1X«.  ----  a,X?ixf.-         -ft  ----  )  =  0. 

This  equation  must  hold  for  all  values  of  x±,  x2,  etc. 

Now  differentiate  successively  partially  with  respect  to  x±,  x2, 
etc.,  and  after  the  differentiation  put  all  the  x's  equal  to  1.  Then  we 
obtain  the  following  set  of  equations  : 


THE  H  THEOREM  39 


Consider  now  the  first  of  these  equations.  Introduce  new  inde- 
pendent variables 

j_  j_ 

a"  =  a%  ft"  —  ft?  i,  -       -  -  etc. 

The  equation  now  becomes 

^*+  +  0r/ «*+:„.«. a 

do"  3/2" 

The  solution  of  this  equation  is  well  known,  it  being  a  special  case 
covered  by  Euler's  theorem.  The  solution  is  the  most  general  homo- 
geneous function  of  the  zeroth  order  in  the  variables  a",  /?", 

etc.  Now  by  a  function  homogeneous  of  the  zeroth  order  we  mean 
such  a  function  that  its  numerical  value  is  unchanged  when  all  the 
arguments  are  multiplied  by  the  same  constant  factor,  which  may 
be  entirely  arbitrary.  It  follows  therefore  that  the  arguments  of  the 

homogeneous  function  must  be  the  original  arguments  a",  ft", 

etc.,  grouped  in  such  a  way  that  they  are  unchanged  in  value  when 
all  of  the  original  arguments  are  multiplied  by  any  factor,  the  same 
for  all.  Now  the  analysis  of  Chapter  II  applies  here  immediately. 
The  analysis  of  that  chapter  applied  to  the  case  of  each  of  the  fun- 
damental units  being  multiplied  by  a  different  arbitrary  factor. 
The  case  here  is  a  special  form  of  the  previous  case,  since  we  here 
require  that  the  result  shall  be  unchanged  when  each  of  the  argu- 
ments is  multiplied  by  the  same  arbitrary  factor.  It  follows  that  the 

most  general  way  in  which  the  arguments  a",  ft", etc.,  can 

be  grouped  is  as  products  of  powers.  If  we  substitute  the  special 
requirements  into  such  a  product  of  powers,  we  see  at  once  that  the 

sum  of  the  exponents  of  a",  ft", etc.,  in  any  such  product  of 

powers  must  be  zero.  Let  us  write  such  a  product  in  the  form 

(a")".   .    (j8")".    • 

Then  we  have  the  condition 


40  DIMENSIONAL  ANALYSIS 

Substituting  now  the  values  of  a,  /?,  ----  etc.,  in  terms  of  a 
----  etc.,  the  typical  product  takes  the  form 


It  is  at  once  obvious  that  there  are  n-1  independent  products  of 
this  form,  since  there  are  n  quantities  a±,  b±,  ----  etc.,  and  these 
are  subject  to  only  one  restriction.  The  arbitrary  function  has,  there- 
fore, n-1  independent  arguments. 

ai 

Consider  now  the  dimensions  of  the  product  a^  ----  in  the 
fundamental  unit  m^  Since  the  dimensions  of  a  in  m±  are  al5  the 

ai  bi  f 

dimensions  of  a«!  in  n^  are  ax.  Similarly  the  dimensions  of  /?  ^  inn^ 
areb-t.But'the  exponents  satisfy  the  condition  a±  +  b±  -]  -----  =  0, 
so  that  we  see  at  once  that  the  products  which  are  the  arguments  of 
the  arbitrary  function  must  all  be  dimensionless  in  n^. 

The  second  of  the  equations  A,  by  precisely  the  same  argument 
imposes  the  additional  restriction  that  the  arguments  of  the  arbi- 
trary function  be  products  of  the  arguments  which  are  dimension- 
less  in  m2.  In  imposing  this  additional  restriction,  the  number  of 
independent  arguments  of  the  arbitrary  function  is  diminished 
from  n  —  1  to  n  —  2. 

In  the  same  way  the  remaining  equations  of  A  demand  that  the 
products  shall  be  dimensionless  in  m3,  m4,  etc.,  until  finally  we  find 
that  the  products  must  be  dimensionless  in  all  the  fundamental 
units.  At  the  same  time,  the  number  of  independent  products,  which 
serve  as  the  arguments  of  the  unknown  function,  are  cut  down  in 
number  to  n—  m. 

Hence  we  have  the  final  result.  //  the  equation  <f>  (a,  /?,  y,  ----  ) 
=  0  is  to  be  a  complete  equation,  the  solution  has  the  form 

F(n1?n2,  ----  )=o 

where  the  n's  are  the  n—m  independent  products  of  the  arguments 
a,  /?,  ----  et  c.,  which  are  dimensionless  in  the  fundamental  units. 
The  result  stated  in  this  form  is  known  as  the  H  theorem,  and 
seems  to  have  been  first  explicitly  stated  by  Buckingham,4  although 
an  equivalent  result  had  been  used  by  Jeans,3  without  so  explicit  a 
statement. 


THE  H  THEOREM  41 

The  solution  in  the  form  above  may  be  solved  explicitly  for  any 
one  of  the  products,  giving  the  equivalent  form  of  result 

a  =  p**-f* 4>(n2,n3,-      -) 

where  the  x  's  are  such  that  a  ft~xi  y~*2 is  dimensionless. 

The  result  in  this  form  embodies  the  mathematical  statement  of 
the  principle  of  dimensional  homogeneity.  For  the  arbitrary  func- 
tion on  the  right-hand  side  is  a  function  of  arguments  each  of  which 
is  of  zero  dimensions,  so  that  every  term  of  the  resulting  function 
must  itself  be  dimensionless.  Every  term  of  this  function  is  to  be 
multiplied  by  a  term  of  the  same  dimensions  as  the  left-hand  side  of 
the  equation,  with  the  result  that  every  term  on  the  right-hand  side 
has  the  same  dimensions  as  the  left-hand  side.  The  terms  may  now 
be  rearranged  in  any  way  that  we  please,  but  whatever  the  rear- 
rangement, the  dimensions  of  all  terms  will  remain  the  same.  This 
is  known  as  the  principle  of  dimensional  homogeneity. 

The  attempt  is  often  made  to  give  an  off-hand  proof  of  the  princi- 
ple of  dimensional  homogeneity  from  the  point  of  view  which 
regards  a  dimensional  formula  as  an  expression  of  the  "essential 
physical  nature"  of  a  quantity.  Thus  it  is  said  that  an  equation 
which  is  an  adequate  expression  of  the  physical  facts  must  remain 
true  no  matter  how  the  fundamental  units  are  changed  in  size,  for 
a  physical  relationship  cannot  be  dependent  on  an  arbitrary  choice 
of  units,  and  if  the  equation  is  to  remain  true  for  all  choices  of  units 
the  dimensions  of  each  term  must  be  the  same,  for  otherwise  we 
would  have  quantities  of  different  physical  natures  put  equal  to 
each  other.  For  instance,  we  could  not  according  to  this  view  have 
a  quantity  of  the  dimensions  of  a  length  on  the  one  side  of  an  equa- 
tion equal  to  a  quantity  of  the  dimensions  of  an  area  on  the  other 
side,  for  it  is  absurd  that  an  area  should  be  equal  to  a  length.2  The 
criticism  of  this  point  of  view  should  be  obvious  after  what  has 
been  said  about  an  equation  merely  being  an  equation  between 
numbers  which  are  the  numerical  measures  of  certain  physical 
quantities. 

It  is  to  be  most  carefully  noticed  that  the  work  above  was  subject 
to  a  most  important  tacit  restriction  at  the  very  outset.  In  putting 
<£  (a,  (3, )  =  0  it  was  tacitly  assumed  that  this  is  the  only  rela- 
tion between  a,  ft etc.,  and  that  the  partial  derivatives  may 

be  computed  in  the  regular  way  on  this  assumption.  If  a,  ft  y,  etc., 
are  connected  by  other  relations  than  </>  (a,  ft )  =  0,  then  the 


42  DIMENSIONAL  ANALYSIS 

analysis  above  does  not  hold,  and  the  results  are  no  longer  true.  For 
it  is  not  true  in  general  that  an  equation  which  is  a  complete  equa- 
tion, that  is,  an  equation  which  remains  true  when  the  size  of  the 
fundamental  units  is  changed,  is  dimensionally  homogeneous.  Such 
an  equation  is  dimensionally  homogeneous  of  necessity  only  when 
there  is  no  other  numerical  relation  between  the  variables  than  that 
defined  by  the  equation  itself.  Consider  as  an  example  a  falling 
body.  Let  v  be  its  velocity,  s  the  distance  of  fall,  t  the  time  of  fall, 
and  g  the  acceleration  of  gravity.  Now  these  quantities  are  related, 
and  there  is  more  than  one  equation  of  connection,  because  both  v 
and  s  are  fixed  when  t  and  g  are  given.  The  relations  connecting 
these  quantities  are  v  =  gt,  and  s  =  %gt2.  In  the  light  of  the  above 
we  would  expect  that  a  complete  equation  connecting  v,  s,  g,  and  t 
need  not  be  dimensionally  homogeneous.  An  example  can  be  given 
immediately,  namely, 

v  +  s  =  gt  +  y2gt2. 

This  is  obviously  a  complete  equation  in  that  it  is  true  and  remains 
true  no  matter  how  the  fundamental  units  of  length  and  time  are 
changed  in  size.  We  may,  if  we  please,  write  from  these  elements 
an  equation  which  is  very  much  more  unusual  and  offensive  in 
appearance,  such  as 

r   .       S  -f  gtl  sinh(s-*gt2) 

v    sin  — — —  =  gt  cosh  (v  —  gt). 

L         v    J 

This  again  is  a  complete  equation;  it  is  not  dimensionally  homo- 
geneous, and  also  offends  our  preconceived  notions  of  what  is  possi- 
ble in  the  way  of  transcendental  functions. 

The  possibility  of  equations  like  those  just  considered  is  in  itself 
a  refutation  of  the  intuitional  method  of  proof  of  the  principle  of 
dimensional  homogeneity  sometimes  given. 

The  equation  v  +  s  =  gt  +  %gt2  reminds  one  of  the  procedure 
used  in  vector  analysis,  in  which  three  scalar  equations  may  be 
replaced  by  a  single  vector  equation.  Obviously  we  may  add  together 
any  number  of  complete  equations  and  obtain  a  result  which  remains 
true.  And  provided  that  the  dimensions  of  the  original  equations 
were  all  different,  the  resulting  compound  equation  (complete  but 
not  dimensionally  homogeneous)  may  be  decomposed,  like  the  vector 
equation,  into  a  number  of  simpler  equations,  by  picking  out  the 
parts  with  the  same  dimensions.  I  do  not  know  whether  this  method 


THE  H  THEOREM 


43 


of  throwing  the  results  into  a  compact  form  can  ever  be  made  to 
yield  any  practical  advantages  or  not. 

Let  us  now  return  to  the  first  form  in  which  we  put  the  result 
above,  namely, 

FCn^n,,-      -)=0. 

Consider  the  n  's  and  how  they  are  formed  from  the  variables.  "Write 
a  typical  n  in  the  form 


The  a,  b,  c,  etc.,  are  to  be  so  chosen  that  this  is  dimensionless. 
Substituting  now  the  dimensional  symbols  of  a,  /?,  ----  etc.,  gives 
as  many  equations  of  condition  between  a,  b,  c,  etc.,  as  there  are 
kinds  of  fundamental  unit.  The  equations  are 


a2a  + 

I 


=  0 


There  are  m  equations,  each  with  n  terms.  Now  the  theory  of  the 
solutions  of  such  sets  of  equations  may  be  found  in  any  standard 
work  on  algebra.  In  general,  n  will  be  greater  than  m.  Under  these 
conditions  there  will  in  general  be  n— m  independent  sets  of  solu- 
tions. That  is,  there  will  in  general  be  n— m  independent  dimension- 
less  products,  and  the  arbitrary  function  F  will  be  a  function  of 
n— m  variables. 

In  certain  special  cases  this  conclusion  will  have  to  be  modified. 
If,  for  instance,  n  =  m,  there  will  in  general  be  no  solution,  but 
there  may  be  in  the  special  case  that  the  determinant  of  the  ex- 
ponents 


vanishes. 

Furthermore,  there  may  be  more  than  n— m  independent  solutions 
if  it  should  happen  that  all  the  m-rowed  determinants  of  the  ex- 
ponents vanish.  This,  of  course,  will  not  very  often  occur,  but  we 
shall  meet  at  least  one  example  later. 


44  DIMENSIONAL  ANALYSIS 

In  the  general  case,  where  there  are  n— m  independent  solutions, 
it  is  generally  possible  to  select  n— m  of  the  quantities  a,  b,  c,  etc.,  in 
any  convenient  way,  assign  to  them  n— m  sets  of  independent  values, 
and  solve  for  the  remaining  quantities,  thus  obtaining  n— m  sets  of 
values  which  determine  n— m  dimensionless  products.  Sometimes 
this  is  not  possible,  and  the  particular  set  of  the  quantities  a,  b,  c, 
etc.,  to  which  arbitrary  values  can  be  assigned  cannot  be  chosen  with 
complete  freedom.  This  occurs  when  certain  determinants  chosen 
from  the  array  of  the  exponents  vanish.  We  will  not  stop  here  to 
develop  a  general  theory,  but  let  the  exceptions  take  care  of  them- 
selves, as  it  is  always  easy  to  do  in  any  special  problem. 

It  is  to  be  noticed  that  the  n  theorem  does  not  contain  anything 
essentially  new,  and  does  not  enable  us  to  treat  any  problems  which 
we  could  not  already  have  handled  by  the  methods  of  the  introduc- 
tion. The  advantage  of  the  theorem  is  one  of  convenience ;  it  places 
the  result  in  a  form  in  which  it  can  be  used  with  little  mental  effort, 
and  in  a  form  of  a  good  deal  of  flexibility,  so  that  the  results  of  the 
dimensional  analysis  may  be  exhibited  in  a  variety  of  forms,  de- 
pending on  the  variables  in  which  we  are  particularly  interested.  In 
this  way  it  has  very -important  advantages. 

The  result  of  this  dimensional  analysis  places  no  restrictions 
whatever  on  the  form  of  the  functions  by  which  the  results  of  experi- 
ments may  be  expressed,  but  the  restriction  is  on  the  form  of  the 
arguments  only.  However  complicated  the  function,  if  it  is  one 
which  satisfies  the  fundamental  requirements  of  the  theory  as  de- 
veloped above,  it  must  be  possible  to  rearrange  the  terms  in  such  a 
way  that  it  appears  as  a  function  of  dimensionless  arguments  only. 
Now  in  using  the  theorem  we  are  nearly  always  interested  in  ex- 
pressing one  of  the  quantities  as  a  function  of  the  others.  This  is 
done  by  solving  the  function  for  the  particular  dimensionless 
product  in  which  the  variable  in  question  is  located,  and  then 
multiplying  that  dimensionless  product  (and  of  course  the  other 
side  of  the  equation  as  well)  by  the  reciprocal  of  the  other  quanti- 
ties which  are  associated  with  it  in  the  dimensionless  product.  The 
result  is  that  on  the  one  side  of  the  equation  the  variable  stands 
alone,  while  on  the  other  side  is  a  product  of  certain  powers  of  some 
of  the  other  variables  multiplied  into  an  arbitrary  function  of  the 
other  dimensionless  products.  This  arbitrary  function  may  be  tran- 
scendental to  the  worst  degree ;  there  is  absolutely  no  restriction  on 
it,  but  its  arguments  are  dimensionless.  This  agrees  with  the  result 


THE  H  THEOREM  45 

of  common  experience  in  regard  to  the  nature  of  the  possible  func- 
tional relations.  We  have  come  to  expect  that  any  argument  which 
appears  under  the  sign  of  a  transcendental  function  must  be  a 
dimensionless  argument.  This  is  usually  expressed  by  saying  that  it 
makes  no  sense  to  take  the  hyperbolic  sine,  for  example,  of  a  time, 
but  the  only  thing  of  which  we  can  take  the  sinh  is  a  number.5  Now 
although  the  observation  is  correct  which  remarks  that  the  argu- 
ments of  the  sinh  functions  which  appear  in  our  analysis  are  usually 
dimensionless,  the  reason  assigned  for  it  is  not  correct.  There  is  no 
reason  why  we  should  not  take  the  sinh  of  the  number  which  meas- 
ures a  certain  interval  of  time  in  hours,  any  more  than  we  should 
not  take  the  number  which  counts  the  number  of  apples  in  a  peck. 
Both  operations  are  equally  intelligible,  but  the  restrictions  imposed 
by  the  II  theorem  are  such  that  we  seldom  see  written  the  sinh  of  a 
dimensional  quantity,  and  even  if  we  should,  it  would  be  possible  by 
a  rearrangement  of  terms,  as  already  explained,  to  get  rid  of  the 
transcendental  function  of  the  dimensional  argument  by  coalescing 
two  or  more  such  functions  into  a  sinh  of  a  single  dimensionless 
argument.  Thus  it  is  perfectly  correct  to  write  the  equation  of  a 
falling  body  in  the  form 

sinh  v  =  sinh  gt, 

but  no  one  would  do  it,  because  this  form  is  more  complicated  than 
that  obtained  by  taking  the  sinh"1  of  both  sides.  The  equation  above 
might  be  rewritten 

sinh  v  cosh  gt  —  cosh  v  sinh  gt  —  0, 

in  which  form  the  rearrangement  to  get  rid  of  the  transcendental 
function  of  a  dimensional  argument  is  not  so  immediate,  particu- 
larly if  one  is  rusty  on  his  trigonometric  formulas.  But  the  last 
form  is  perfectly  adapted  for  numerical  computation,  in  the  sense 
that  it  will  always  give  the  correct  result,  and  still  holds  when  the 
size  of  the  fundamental  units  is  changed. 

There  is  a  corollary  to  these  remarks  about  transcendental  func- 
tions with  respect  to  the  exponents  of  powers.  It  is  obvious  that  in 
general  we  cannot  have  an  exponent  which  has  dimensions.  If  such 
appears,  it  is  possible  to  combine  it  with  others  in  such  a  way  that 
the  dimensionality  is  lost.  But  there  is  absolutely  no  restriction 
whatever  imposed  as  to  numerical  exponents ;  these  may  be  integral, 
or  fractional,  or  incommensurable.  It  is  often  felt  that  the  dimen- 


46  DIMENSIONAL  ANALYSIS 

sional  formula  of  a  quantity  should  not  involve  the  fundamental 
quantities  to  fractional  powers.6  This  is  a  part  of  the  view  that 
regards  a  dimensional  formula  as  an  expression  of  operations  on 
concrete  physical  things,  and  this  point  of  view  finds  it  hard  to 
assign  a  meaning  to  the  two-thirds  power  of  a  time,  for  example. 
But  it  seems  to  me  just  as  hard  to  assign  a  physical  meaning  to  a 
minus  second  power  of  a  time,  and  the  possibility  of  such  exponents 
is  admitted  by  everyone. 

The  II  theorem  as  given  contains  all  the  elements  of  the  situation. 
But  in  use  there  is  a  great  deal  of  flexibility  in  the  choice  of  the 
arguments  of  the  function,  as  is  suggested  by  the  fact  that  it  is 
possible  to  choose  the  independent  solutions  of  a  set  of  algebraic 
equations  in  a  great  number  of  ways.  The  way  in  which  the  inde- 
pendent solutions  are  chosen  determines  the  form  of  the  dimension- 
less  products,  and  the  best  form  for  these  will  depend  on  the  par- 
ticular problem.  We  shall  in  chapter  VI  treat  a  number  of  concrete 
examples  which  will  illustrate  how  the  products  are  to  be  chosen  in 
special  cases. 

REFEEENCES 

(1)  E.  Buckingham,  Phys.  Rev.  4,  345,  1914. 

(2)  Routh. 

(3)  J.  H.  Jeans,  Proc.  Roy.  Soc.  76,  545,  1905. 

(4)  E.  Buckingham,  Reference  1,  also  Jour.  Wash.  Acad.  Sci.  4, 
347,  1914. 

(5)  E.  Buckingham,  Ref.  1,  page  346. 

' '  Such  expressions  as  log  Q  or  sin  Q  do  not  occur  in  physical 
equations ;  for  no  purely  arithmetical  operator,  except  a  simple 
numerical  multiplier,  can  be  applied  to  an  operand  which  is  not 
itself  a  dimensionless  number,  because  we  cannot  assign  any 
definite  meaning  to  the  result  of  such  an  operation. ' ' 

See  also  in  this  connection  page  266  of  the  following. 

(6)  S.  P.   Thompson,   Elementary  Lessons  in  Electricity   and 
Magnetism,  p.  352. 

' '  It  also  seems  absurd  that  the  dimensions  of  a  unit  of  elec- 
tricity should  have  fractional  powers,  since  such  quantities  as 
M»  and  L*  are  meaningless. ' ' 

W.  Williams,  Phil.  Mag.  34,  234,  1892. 

' '  So  long,  however,  as  L,  M,  and  T  are  fundamental  units,  we 
cannot  expect  fractional  powers  to  occur.  .  .  .  Now  all  dynami- 
cal conceptions  are  built  up  ultimately  in  terms  of  these  three 


THE  n  THEOREM  47 

ideas,  mass,  length,  and  time,  and  since  the  process  is  syntheti- 
cal, building  up  the  complex  from  the  simple,  it  becomes  ex- 
pressed in  conformity  with  the  principles  of  Algebra  by  integral 
powers  of  L,  M,  and  T.  .  .  .  Obviously  if  mass,  length,  and  time 
are  to  be  ultimate  physical  conceptions,  we  cannot  give  inter- 
pretations to  fractional  powers  of  L,  M,  and  T,  because  we  can- 
not analyze  the  corresponding  ideas  to  anything  simpler.  We 
should  thus  be  unable,  according  to  any  physical  theory,  to 
give  interpretations  to  formulae  involving  fractional  powers 
of  the  fundamental  units. ' ' 


CHAPTER  V 

DIMENSIONAL  CONSTANTS  AND  THE  NUMBER 
OF  FUNDAMENTAL  UNITS 

THE  essential  result  which  we  have  obtained  in  the  II  theorem  is  in 
the  restriction  which  it  places  on  the  number  of  arguments  of  the 
arbitrary  function.  The  fewer  the  arguments,  the  more  restricted 
the  function,  and  the  greater  our  information  about  the  answer. 
Thus  if  the  problem  is  such  that  there  are  four  variables,  and  three 
fundamental  kinds  of  unit,  our  analysis  shows  that  there  is  only  one 
dimensionless  product,  which  we  can  determine,  and  that  some  func- 
tion of  this  product  is  zero.  This  is  equivalent  to  saying,  in  this 
special  case,  that  the  product  itself  is  some  constant,  and  we  have 
complete  information  as  to  the  nature  of  the  solution,  except  for  the 
numerical  value  of  the  constant.  This  was  the  nature  of  the  solution 
which  we  found  for  the  pendulum  problem.  If  it  had  not  been  for 
the  dimensional  analysis,  any  conceivable  relation  between  the  four 
arguments  might  have  been  possible,  and  we  should  have  had  abso- 
lutely no  information  about  the  solution.  Similarly,  if  there  are  two 
more  variables  than  fundamental  kind  of  quantity,  there  will  be 
two  dimensionless  products.  The  solution  is  an  arbitrary  function 
of  these  two  products  put  equal  to  zero,  which  may  be  solved  for  one 
of  the  products  as  a  function  of  the  other.  This  was  the  case  with  the 
heat  transfer  problem  already  treated.  It  certainly  gives  more  infor- 
mation to  know  that  the  solution  is  of  this  form  than  merely  to 
know  that  there  is  some  function  of  the  five  variables  which  van- 
ishes, which  was  all  that  we  could  say  before  we  applied  our 
analysis. 

It  is  to  our  advantage,  evidently,  that  the  number  of  arguments 
which  are  to  be  connected  by  the  functional  relation  should  be  as 
small  as  possible.  Now  the  variables  which  enter  the  functional  rela- 
tion to  which  our  analysis  has  been  applied  comprise  all  the  varia- 
bles which  can  change  in  numerical  magnitude  under  the  conditions 
of  the  problem.  These  variables  are  of  two  kinds.  First  are  the 


DIMENSIONAL  CONSTANTS  49 

physical  variables,  which  are  the  measures  of  certain  physical  quan- 
tities, and  which  may  change  in  magnitude  over  the  domain  to 
which  our  result  is  to  apply.  The  numbers  measuring  these  physical 
quantities  may  also  change  when  the  size  of  the  fundamental  units 
changes.  In  the  second  place,  there  may  be  other  arguments  of  the 
nature  of  coefficients  in  the  equation  which  do  not  change  in  numeri- 
cal magnitude  when  the  physical  system  alone  changes,  but  which 
change  in  magnitude  when  the  size  of  the  fundamental  measuring 
units  changes.  It  is  these  which  we  have  called  dimensional  con- 
stants. Now  in  any  actual  case  we  are  interested  only  in  the  physical 
problem,  and  are  interested  in  finding  a  relation  between  the  physi- 
cally variable  quantities.  The  dimensional  constants  are  to  be 
regarded  as  an  evil,  to  be  tolerated  only  if  they  make  possible  more 
information  about  the  physical  variables. 

"We  thus  see  that  the  n  theorem  applies  to  the  aggregate  of  physi- 
cal variables  and  dimensional  constants,  whereas  we  are  interested 
primarily  in  the  physical  variables  alone.  If  the  number  of  dimen- 
sional constants  is  so  great  that  the  number  of  arguments  of  the 
arbitrary  function  allowed  by  the  II  theorem  is  equal  to  or  greater 
than  the  number  of  physical  variables  alone,  then  we  are  no  better 
off  after  applying  our  n  theorem  than  before.  Now  we  have  already 
seen  that  in  the  worst  possible  case  the  number  of  dimensional  con- 
stants cannot  exceed  the  number  of  physical  variables,  for  any 
empirical  equation  can  be  made  complete  by  the  introduction  of  a 
dimensional  constant  with  each  physical  variable.  Furthermore,  it 
is  almost  always  true  that  the  number  of  physical  variables  is  equal 
to  or  greater  than  the  number  of  fundamental  units.  Hence,  if  the 
number  of  dimensional  constants  is  equal  the  number  of  physical 
variables,  the  number  of  dimensionless  products  is  greater  than  or 
at  most  equal  to  the  number  of  physical  variables.  In  the  general 
case,  therefore,  the  II  theorem  gives  no  new  information.  Hence  it 
is  of  the  utmost  importance  to  keep  down  to  the  minimum  the  num- 
ber of  dimensional  constants  used  in  the  equation. 

When,  therefore,  shall  we  expect  dimensional  constants,  and  in 
any  particular  problem  how  shall  we  find  what  they  are,  and  what 
are  their  dimensional  formulas?  The  answer  to  this  question  is 
closely  related  to  the  answer  to  the  question  of  how  we  shall  choose 
the  list  of  physical  quantities  between  which  we  are  to  search  for  a 
relation.  We  have  seen  that  it  does  not  do  to  merely  ask  ourselves 
' l  Does  the  result  depend  on  this  or  that  physical  quantity  ? ' '  for  we 


50  DIMENSIONAL  ANALYSIS 

have  seen  in  one  problem  that  although  the  result  certainly  does 
"depend"  on  the  action  of  the  atomic  forces,  yet  we  do  not  have 
to  consider  the  atomic  forces  in  our  analysis,  and  they  do  not  enter 
the  functional  relation. 

To  answer  the  question  of  what  variables  to  include  demands  a 
background  of  a  great  deal  of  physical  experience.  If  we  are  to  treat 
a  certain  problem  by  the  methods  of  mechanics  we  must  have  enough 
background  to  be  assured  that  the  problem  is  a  problem  in  mechan- 
ics, and  involves  essentially  no  elements  that  are  not  treatable  by  the 
ordinary  equations  of  mechanics.  We  must  know  that  certain 
aspects  of  the  problem  can  be  neglected,  and  that  certain  others 
alone  are  essential  as  far  as  certain  features  of  behavior  go.  No  one 
would  say  that  in  any  problem  of  mechanics  the  atomic  forces  are 
not  essential,  but  our  experience  shows  that  they  combine  into 
certain  complexes,  which  may  be  sufficiently  characterized  by  an 
analysis  which  does  not  go  down  to  the  ultimate  component  parts, 
and  that  the  results  of  our  analysis,  which  disregards  many  even 
essential  aspects  of  the  situation,  have  validity  under  certain  condi- 
tions whose  restrictions  are  not  irksome.  The  experience  involved  in 
judgments  of  this  sort  reaches  so  far  back  that  we  know  almost  by 
instinct  whether  a  problem  is  suitable  for  mechanical  treatment  or 
not.  And  if  the  problem  is  capable  of  mechanical  treatment,  we 
know,  by  the  very  definition  of  what  we  mean  by  a  mechanical  sys- 
tem, what  the  equations  are  which  the  motion  of  the  component 
parts  of  the  system  conform  to,  and  what  the  form  of  the  equations 
is.  In  the  same  way,  we  know  by  instinct  whether  a  system  is  a 
thermodynamic  system,  or  an  electrical  system,  or  a  chemical  sys- 
tem, and  in  each  case,  because  we  know  what  we  mean  when  we  say 
that  a  phenomenon  is  of  such  or  such  a  nature,  we  know  what  are 
the  laws  which  govern  the  variations  of  the  system,  and  the  ele- 
ments which  must  be  considered  in  formulating  the  relations  be- 
tween the  parts.  But  a  very  wide  background  of  experience,  extend- 
ing over  many  generations,  was  necessary  before  we  could  say  that 
this  particular  group  of  phenomena  is  mechanical  or  electrical,  or, 
in  general,  that  the  phenomenon  is  physical. 

Now  my  point  of  view  is  essentially  that  precisely  the  same  ex- 
perience which  is  demanded  to  enable  us  to  say  whether  a  system 
is  mechanical  or  electrical  is  the  experience  which  is  demanded  in 
order  to  enable  us  to  make  a  dimensional  analysis.  This  experience 
will  in  the  first  place  inform  us  what  physical  variables  to  include 


DIMENSIONAL  CONSTANTS  51 

in  our  list,  and  will  in  the  second  place  tell  us  what  dimensional 
constants  are  demanded  in  any  particular  problem. 

Let  us  for  the  present  forget  what  we  know  of  dimensional  analysis 
and  imagine  ourselves  approaching  a  new  problem.  In  the  first  place 
we  decide  in  the  light  of  the  experience  of  all  the  ages  what  the 
nature  of  the  problem  is.  Suppose  that  we  decide  that  it  is  mechani- 
cal. Then  we  know  that  the  motion  of  the  system  is  governed  by  the 
laws  of  mechanics,  and  we  know  what  these  laws  are.  We  write  down 
certain  equations  of  motion  of  the  system.  We  are  careful  to  include 
all  the  equations  of  motion,  so  that  the  system  of  equations  by  which 
we  have  described  the  relations  between  the  parts  of  the  system  has 
a  unique  solution.  Then  we  are  convinced,  because  of  our  past 
experience,  that  we  have  essentially  represented  all  the  elements  of 
the  situation,  that  our  equations  correspond  to  the  reality  at  least  as 
far  as  certain  aspects  of  the  phenomenon  go,  and  the  solution  of  the 
equations  will  correctly  represent  the  behavior  of  the  system  which 
we  have  thus  analyzed.  We  are  not  disappointed.  The  fact  that  our 
predictions  turn  out  to  be  verified  means  merely  that  we  have 
become  masters  of  a  certain  group  of  natural  phenomena. 

Now  the  astute  observer  (Fourier1  was  the  first  astute  observer) 
notices  that  the  equations  by  which  the  relation  of  the  component 
parts  of  the  system  is  analyzed  are  expressed  in  such  a  general  form 
that  they  remain  true  when  the  size  of  the  fundamental  units  is 
changed.  For  instance,  the  equation  stating  that  the  force  acting  on 
a  particular  part  of  our  mechanical  system  is  equal  to  the  mass  of 
that  part  times  its  acceleration  remains  true  however  the  size  of  the 
fundamental  units  is  changed,  because  in  every  system  of  units 
which  we  use  for  mechanical  purposes,  the  unit  of  force  is  defined 
so  that  force  has  this  relation  to  mass  and  acceleration.  Every  one 
of  the  fundamental  equations  of  motion  is  in  the  same  way  a  com- 
plete equation.  The  final  solution  is  obtained  from  the  equations  of 
motion  by  a  purely  mathematical  process,  which  has  no  relation  to 
the  size  of  the  fundamental  units.  It  follows,  therefore,  in  general, 
that  the  final  result  will  also  be  complete,  in  the  sense  that  the  equa- 
tion expressing  the  final  result  is  a  complete  equation. 

Dimensional  analysis  may,  therefore,  be  applied  to  the  results 
which  we  obtain  by  solving  the  equations  of  motion.  (We  use  equa- 
tions of  motion  in  a  general  sense,  applying  to  thermodynamic  and 
electrical  as  well  as  mechanical  systems. )  Now  the  arguments  of  the 
function  which  we  finally  obtain  by  solving  the  equations  of  motion 


52  DIMENSIONAL  ANALYSIS 

can  obviously  be  only  those  quantities  which  we  put  into  the  original 
equations  of  motion,  for  the  mathematical  operations  can  introduce 
no  new  arguments. 

In  particular,  the  dimensional  constants  which  enter  the  final  relation 
are  those,  and  those  only,  which  we  had  to  use  in  writing  down  the  equa- 
tions of  motion.  This  is  the  entire  essence  of  the  question  of  dimensional 
constants. 

With  regard  to  the  dimensional  formulas  of  dimensional  con- 
stants, we  may  merely  appeal  to  experience  with  the  observation 
that  all  such  constants  are  of  the  form  of  products  of  powers  of  the 
fundamental  quantities.  But  it  is  evident  on  reflection,  that  any  law 
of  nature  can  be  expressed  in  a  form  in  which  the  dimensional 
formulas  of  the  constants  are  of  this  type,  by  the  device,  already 
adopted,  of  introducing  dimensional  constants  as  factors  with  the 
measured  quantities  in  such  a  way  as  to  make  the  equation  com- 
plete. We  will  hereafter  assume  that  the  equations  of  motion  (which 
are  merely  expressions  of  the  laws  of  nature  governing  phenomena) 
are  thrown  into  such  a  form  that  the  dimensional  constants  are  of 
this  type ;  this  is  seen  to  involve  no  real  restriction. 

It  appears,  therefore,  that  dimensional  analysis  is  essentially  of 
the  nature  of  an  analysis  of  an  analysis.  We  must  know  enough 
about  the  situation  to  know  what  the  general  nature  of  the  problem 
is,  and  what  the  elements  are  which  would  be  introduced  in  writing 
down  the  equations  determining  the  motion  (in  the  general  sense) 
of  the  system.  Then,  knowing  the  nature  of  the  elements,  we  can 
obtain  certain  information  about  the  necessary  properties  of  any 
relations  which  can  be  deduced  by  mathematical  manipulations 
with  the  elements.  In  so  far  as  our  knowledge  of  the  underlying 
laws  of  nature  is  adequate  we  may  have  confidence  in  the  result, 
but  the  result  can  have  no  validity  not  pertaining  to  the  equations 
of  motion,  and  is  in  no  way  different  from  all  our  other  knowledge. 
The  result  is  approximate,  as  the  laws  of  motion  are  approximate,  a 
restriction  which  is  imposed  by  the  very  nature  of  knowledge  itself. 

The  man  applying  dimensional  analysis  is  not  to  ask  himself  ' '  On 
what  quantities  does  the  result  depend?"  for  this  question  gets 
nowhere,  and  is  not  pertinent.  Instead  we  are  to  imagine  ourselves 
as  writing  out  the  equations  of  motion  at  least  in  sufficient  detail  to 
be  able  to  enumerate  the  elements  which  enter  them.  It  is  not  neces- 
sary to  actually  write  down  the  equations,  still  less  to  solve  them. 


DIMENSIONAL  CONSTANTS  53 

Dimensional  analysis  then  gives  certain  information  about  the 
necessary  character  of  the  results.  It  is  here  of  course  that  the 
advantage  of  the  method  lies,  for  the  results  are  applicable  to  sys- 
tems so  complicated  that  it  would  not  be  possible  to  write  the  equa- 
tions of  motion  in  detail. 

It  is  to  be  especially  noticed  that  the  results  of  dimensional 
analysis  cannot  be  applied  to  any  system  whose  fundamental  laws 
have  not  yet  been  formulated  in  a  form  independent  of  the  size  of 
the  fundamental  units.  For  instance,  dimensional  analysis  would 
certainly  not  apply  to  most  of  the  results  of  biological  measure- 
ments, although  such  results  may  perfectly  well  have  entire  physical 
validity  as  descriptions  of  the  phenomena.  It  would  seem  that  at 
present  biological  phenomena  can  be  described  in  complete  equa- 
tions only  with  the  aid  of  as  many  dimensional  constants  as  there 
are  physical  variables.  In  this  case,  we  have  seen,  dimensional  analy- 
sis has  no  information  to  give.  In  a  certain  sense,  the  mastery  of  a 
certain  group  of  natural  phenomena  and  their  formulation  into 
laws  may  be  said  to  be  coextensive  with  the  discovery  of  a  restricted 
group  of  dimensional  constants  adequate  to  coordinate  all  the 
phenomena. 

Let  us  apply  this  view  of  the  nature  of  dimensional  constants  to 
the  problem  which  we  have  already  considered  of  the  electro- 
magnetic mass  of  a  spherical  distribution  of  electricity.  This  is  evi- 
dently a  problem  in  electrodynamics,  and  must  be  solved  by  the  use 
of  the  field  equations.  These  field  equations  consist  of  certain  mathe- 
matical operators  operating  on  certain  combinations  of  the  electric 
and  magnetic  forces  and  the  velocity  of  light.  In  this  particular 
problem  we  want  to  solve  the  equations  in  such  a  form  as  to  get  the 
electromagnetic  mass;  this  is  the  integral  throughout  space  of  a 
constant  times  the  energy  density,  which  in  turn  is  given  by  the 
distribution  of  the  forces,  which  are  determined  by  the  distribution 
of  the  charge.  Hence  if  there  is  a  relation  of  the  form  which  we 
suspect,  the  forces  will  eliminate  from  the  final  result.  There  is, 
however,  no  reason  to  think  that  the  characteristic  constant  "c" 
of  the  equations  will  also  eliminate  from  the  result,  and  we  must 
therefore  seek  for  a  relation  between  the  total  charge,  the  mass,  the 
radius,  and  the  constant  of  the  field  equations,  which  is  the  velocity 
of  light.  This  relation  we  have  already  found,  and  checked  against 
the  results  of  a  detailed  solution  of  the  field  equations  applied  to 
this  particular  problem. 


54  DIMENSIONAL  ANALYSIS 

We  have  seen  that  dimensional  constants  are  going  to  enter  the 
final  result  only  in  so  far  as  they  enter  the  equations  of  motion.  Now 
a  dimensional  constant  in  an  equation  of  motion  is  an  expression  of 
a  physical  relation  which  is  so  universal  as  to  be  characteristic  of  all 
the  phenomena  embraced  in  the  particular  group  which  we  are 
considering.  Such  a  universal  physical  relation  may  be  treated  in 
two  ways.  We  may  leave  the  dimensional  constant  in  the  equations 
as  an  explicit  statement  of  the  relation,  as  i^  done  in  the  field  equa- 
tions of  electrodynamics,  or  we  may  define  our  fundamental  units 
with  this  relation  in  view,  thus  obtaining  a  system  of  units  in  which 
the  dimensional  constant  has  disappeared  but  in  which  the  number 
of  units  which  may  be  regarded  as  fundamental  has  been  restricted 
in  such  a  way  that  all  units  belonging  to  the  system  automatically 
bear  the  experimental  relation  to  each  other.  The  system  of  units  so 
obtained  is  of  value  only  in  treating  that  group  of  phenomena  to 
which  the  law  in  question  applies.  Thus  it  is  a  result  of  experience 
that  the  mass  times  the  acceleration  of  a  body  is  proportional  to  the 
force  acting  upon  it.  In  this  statement  of  the  experimental  facts 
there  is  no  restriction  whatever  upon  the  units  of  mass,  or  length, 
or  time,  or  force.  The  factor  of  proportionality  will  change  in 
numerical  magnitude  whenever  any  one  of  the  four  fundamental 
units  is  changed  in  size.  But  now,  instead  of  being  bothered  by  a 
continually  changing  factor  of  proportionality,  we  may  arbitrarily 
say  that  this  factor  shall  be  unity  in  all  systems  which  we  will  con- 
sider, and  we  will  bring  this  result  about  by  defining  the  unit  of 
force  in  our  new  system  to  be  such  that  the  force  acting  on  a  body 
is  equal  to  the  mass  times  the  acceleration.  We  have  in  this  way 
obtained  a  system  of  units  adapted  to  dealing  with  all  those  physical 
systems  in  which  the  laws  of  motion  involve  a  statement  of  the 
physical  relation  between  force,  mass,  and  acceleration,  but  if  the 
physical  system  should  be  such  that  this  relation  is  not  involved 
in  the  motion  of  the  system,  then  we  would  be  unduly  restricting 
ourselves  by  using  the  mechanical  system  of  units. 

These  considerations^  as  to  the  possible  systems  of  units  answer 
the  question  previously  raised,  as  in  the  fourth  problem  of  the 
introduction,  for  example,  as  to  the  number  of  kinds  of  .units  which 
we  shall  take  as  fundamental.  The  answer  depends  entirely  upon  the 
particular  problem,  and  will  involve  the  physical  relations  which 
are  necessary  to  a  complete  expression  of  the  motion  of  the  parts. 
In  any  ordinary  problem  of  dynamics,  for  example,  the  relation 


DIMENSIONAL  CONSTANTS  55 

between  force  and  mass  and  acceleration  is  essentially  involved  in 
the  equations  of  motion.  This  relation  may  be  brought  into  the 
equations  either  by  the  use  of  four  fundamental  kinds  of  unit, 
force,  mass,  length,  and  time,  with  the  corresponding  dimensional 
constant  of  proportionality,  or  by  using  the  ordinary  mechanical 
units  of  mass,  length,  and  time,  in  which  force  is  defined  so  that  the 
experimental  relationship  is  always  satisfied,  and  the  dimensional 
constant  has  disappeared.  In  either  case  the  results  of  the  dimen- 
sional analysis  are  the  same.  For  the  difference  between  the  number 
of  fundamental  units  and  the  number  of  variables,  which  deter- 
mines the  number  of  arguments  of  the  unknown  function,  is  the 
same  in  either  case,  because  when  the  number  of  units  is  augmented 
by  one  by  including  the  force,  the  number  of  variables  is  also  aug- 
mented by  one  by  including  the  dimensional  constant,  and  the  differ- 
ence remains  constant.  If,  however,  the  problem  were  such  that  the 
experimental  relation  between  force,  mass,  and  acceleration  is  not 
involved  in  the  equations  of  motion  of  the  system,  then  the  ordinary 
mechanical  units  would  be  inappropriate,  because  we  would  obtain 
less  information  when  using  them.  For  we  could  in  this  case  use 
four  fundamental  units  without  introducing  a  corresponding  dimen- 
sional constant  into  the  list  of  variables,  so  that  the  difference  be- 
tween the  number  of  variables  and  the  units  would  be  less  by  one 
when  using  four  than  when  using  three  fundamental  units,  and  the 
arguments  of  the  function  would  be  fewer  in  number,  which  is  de- 
sirable. We  shall  meet  an  example  illustrating  this  point  later. 

EEFEEENCES 

(1)  Fourier,  Theorie  de  Chaleur,  160. 

As  dealing  with  the  general  question  of  the  proper  number  of 
fundamental  units  may  be  mentioned 

E.  Buckingham,  Nat.  96,  208,  and  396,  1915. 


CflAPTER  VI 

EXAMPLES  ILLUSTRATIVE  OF  DIMENSIONAL 
ANALYSIS 

LET  us  in  the  first  place  recapitulate  the  results  of  the  preceding 
chapter.  Before  undertaking  a  dimensional  analysis  we  are  to  im- 
agine ourselves  as  making  an  analysis  to  the  extent  of  deciding  the 
nature  of  the  problem,  and  enumerating  the  physical  variables  which 
would  enter  the  equations  of  motion  (in  the  general  sense)  and 
also  the  dimensional  coefficients  required  in  writing  down  the  equa- 
tions of  motion.  The  dimensions  of  all  these  variables  are  then  to  be 
written  in  terms  of  the  fundamental  units.  These  fundamental  units 
are  to  be  chosen  for  each  particular  problem  in  such  a  way  that 
their  number  is  as  large  as  possible  without  involving  the  introduc- 
tion of  compensating  dimensional  constants  into  the  equations  of 
motion.  The  dimensionless  products  of  the  variables  are  then  to  be 
formed  in  accordance  with  the  II  theorem,  choosing  the  products  in 
such  a  way  from  the  great  variety  possible  that  the  variables  in 
which  we  are  particularly  interested  may  stand  conspicuously  by 
themselves.  Having  formed  the  products,  the  II  theorem  gives  im- 
mediately the  functional  relation. 

In  the  following  illustrative  examples  we  have  particularly  to 
consider  the  proper  number  of  fundamental  units,  and  the  most 
convenient  way  of  choosing  the  dimensionless  products.  The  matter 
of  dimensional  constants  we  regard  as  clear. 

As  the  first  example  we  will  take  the  first  treated  by  Lord  Eay- 
leigh  in  Nature*  Consider  a  wave  advancing  on  deep  water  under 
the  action  of  gravity.  This  is  evidently,  a  problem  in  hydrodynamics, 
which  is  merely  mechanics  applied  to  liquids.  The  equations  of 
mechanics  will  therefore  apply.  Now  the  liquid  when  displaced  from 
equilibrium  is  restored  by  the  force  of  gravity.  This  will  involve  the 
density  of  the  liquid  and  the  intensity  of  gravity.  Evidently  these 
quantities  will  enter  the  equations  of  motion.  No  other  properties 
of  the  liquid,  such  as  the  compressibility,  will  enter,  because  we 
know  from  a  discussion  of  the  equations  of  hydrodynamics  that  such 


EXAMPLES  OF  DIMENSIONAL  ANALYSIS  57 

properties  are  unimportant  for  phenomena  of  this  scale  of  magni- 
tude. Physically,  of  course,  the  compressibility  affects  the  result 
to  a  certain  extent,  so  that  the  result  of  our  analysis  will  not  be 
exact,  but  will  be  a  valid  approximation  only  to  the  extent  that  the 
equations  of  hydrodynamics  are  valid  approximations.  There  are 
no  dimensional  constants  entering  the  equations  of  hydrodynamics, 
provided  that  we  use  ordinary  mechanical  units,  in  which  mass, 
length,  and  time  are  fundamental,  for  the  laws  of  motion  have 
entered  this^  system  of  units  through  the  definition  of  force.  The 
equations,  of  course,  are  equations  between  the  displacements  and 
the  other  elements.  Now  it  is  conceivable  that  we  might  eliminate 
the  various  displacements  from  the  equations,  and  come  out  at  the 
end  with  a  relation  between  the  velocity  of  propagation,  the  density, 
and  the  intensity  of  gravity  only,  analogously  to  the  pendulum 
problem. 

Let  us  try  this.  Write  down  the  variables  and  their  dimensional 
formulas,  as  before. 

Name  of  Quantity.  Symbol.  Dimensional  Formula.* 
Velocity  of  wave,                            v  LT"1 

Density  of  liquid,  d  ML~3 

Accleration  of  gravity,  g  LT~2 

We  now  apply  the  n  theorem.  We  have  three  variables,  and  three 
fundamental  kinds  of  unit.  The  difference  between  these  numbers 
is  zero,  and  therefore,  according  to  the  theorem,  there  are  zero 
dimensionless  products.  That  is,  we  have  made  some  mistake^  and 
no  relation  exists,  unless  this  should  be  one  of  those  exceptional 
cases  in  which  a  product  may  be  formed  of  fewer  than  the  normal 
number  of  factors.  But  an  examination  shows  that  this  is  no  excep- 
tion, and  there  is  in  fact  no  dimensionless  product.  This  shows  that 
the  suggested  elimination  was  not  possible,  but  that  some  other  ele- 
ments or  combination  of  elements  must  enter  the  final  result..  Of 
course  the  detailed  analysis  wilFgive  as  the  final  result  a  detailed 
description  of  the  motion  of  the  water,  from  which  we  must  pick  out 
the  wave  motion  and  find  its  velocity.  That  is,  along  with  the  veloc- 
ity in  the  final  result  there  will  be  something  characteristic  of  the 
particular  wave.  The  velocity  of  all  the  waves  need  not  be  the  same, 
but  may  depend  on  the  wave  length,  for  example.  Physically,  of 
course,  we  knew  this  in  the  beginning,  and  we  were  stupid  only  for 


58  DIMENSIONAL  ANALYSIS 

purposes  of  instruction.  Our  experience  with  problems  of  this 
nature  would  have  led  us  to  search  for  a  relation  between  the  varia- 
bles which  we  put  into  the  analysis,  the  velocity,  and  the  wave 
length.  Let  us  introduce,  therefore,  into  our  list  of  quantities  the 
wave  length. 

Wave  length  A  L 

We  have  now  four  variables,  and  the  dimensional  formulas  are 
expressed  in  terms  of  three  fundamental  kinds  of  unit,  so  that  the 
II  theorem  leads  us  to  expect  the  existence  of  one  dimensionless 
product,  and  the  result  will  be  that  the  dimensionless  product  is 
equal  to  a  constant. 

The  proof  of  the  n  theorem  also  showed  that  one  exponent  in  a 
dimensionless  product  can  be  assigned  arbitrarily.  Since  we  are 
particularly  interested  in  v,  let  us  choose  its  exponent  as  unity,  and 
write  the  dimensionless  product  in  the  form  v  d-tt  g-0  A-*  .  Putting 
this  equal  to  a  constant  and  solving  for  v,  gives  for  the  result 

v  =  Const  da  g^  \y. 

The  dimensions  of  the  factors  on  the  right-hand  side  must 
together  be  the  same  as  that  of  the  velocity,  which  stands  alone  on 
the  left-hand  side.  We  solve  for  the  unknown  exponents  of  the  fac- 
tors of  the  right-hand  side.  Substitute  the  dimensional  formulas  for 
the  variables 

LT-1  =  (ML-8)' 


Now  write  down  in  succession  the  condition  that  the  exponents  of 
M,  of  L,  and  of  T  be  the  same  on  the  two  sides  of  the  equation.  This 
gives 

a  =       0         condition  on  M 

—  3a  +  /?  +  y=       1         condition  on  L 

—  2  (3  =  —  1         condition  on  T. 

Whence 

a  =    0 


7  —  72    J   j 

and  the  final  result  is  of  the  form 

v  ==  Const 

w-f 


EXAMPLES  OF  DIMENSIONAL  ANALYSIS  59 

The  velocity  of  a  gravity  wave  on  deep  water  (the  reason  the 
depth  did  not  enter  the  final  result  was  because  we  postulated  that 
the  water  was  to  be  deep)  is  therefore  proportional  to  the  square 
root  of  the  wave  length  and  the  intensity  of  gravity,  or  is  propor- 
tional to  the  velocity  acquired  by  a  body  falling  freely  under 
gravity  through  a  distance  equal  to  the  wave  length. 

It  is  to  be  noticed  that  the  density  of  the  liquid  has  disappeared 
from  the  final  result.  This  might  have  been  anticipated;  if  the 
density  is  doubled  the  gravitational  force  is  also  doubled  on  every 
element,  and  the  accelerations  and  therefore  all  the  velocities  are 
unaltered,  because  the  doubled  force  is  compensated  by  a  doubled 
mass  of  every  element. 

Since  the  density  disappears  from  the  final  result  we  have  here 
a  dimensionless  product  of  v,  1,  and  g  only.  This  is,  therefore,  a 
dimensionless  product  of  three  variables,  expressed  in  three  funda- 
mental units.  This  is  in  general  not  possible,  but  demands  some 
special  relation  between  the  dimensional  formulas  of  the  variables. 

We  can  see  in  a  moment  by  writing  out  the  product  in  terms  of 
unknown  exponents,  and  then  writing  the  algebraic  equations  which 
the  exponents  must  satisfy,  that  the  condition  that  a  dimensionlegs 
product  exist  in  a  number  of  jterms  just  equal  to  the  number  of 
fundamental  units  ia_that  the  detenfl1TlftTit  of  the  exponents  in  the 
mrnensionarlbrnmla^of  the  factors  vanish.  This  is  obviously  not 
restricted  to  the  case  of  three  fundamental  units,  but  applies  to  any 
number.  Conversely  the  condition  that  a  particular  element  shall 
enter  as  a  factor  into  a  dimensionless  product  with  a  number  of 
other  factors  equal  in  number  to  the  fundamental  units,  is  that  the 
determinant  of  the  exponents  of  the  other  factors  shall  not  be  zero ; 
otherwise  the  other  factors  by  themselves  form  a  dimensionless 
product  into  which  the  factor  in  which  we  are  interested  does  not 
enter. 

Consider  now  a  second  problem.  An  elastic  pendulum  is  made 
by  attaching  to  a  weightless  spring  of  elastic  constant  k  a  box  of 
volume  V  which  is  filled  with  a  liquid  of  density  d.  The  mass  of  the 
liquid  in  the  box  is  acted  upon  by  gravity,  and  we  are  required  to 
find  an  expression  for  the  time  of  oscillation.  As  before  we  make 
a  list  of  the  quantities  and  their  dimensions. 


60  DIMENSIONAL  ANALYSIS 

Name  of  Quantity.  Symbol.  Dimensional  Formula. 
Elastic  constant  (force  per  unit 

displacement),  k-  MT~2 

Time  of  oscillation,  t   -  T 

Volume  of  box,  v  L3 

Density  of  liquid,  d  ML~3 

Acceleration  of  gravity,  g  LT~2 

The  problem  is  obviously  one  in  ordinary  mechanics,  so  that  we 
are  justified  in  using  the  mechanical  system  of  units,  and  there  will 
be  no  dimensional  constant.  The  variables  which  we  have  listed 
above  are,  therefore,  the  only  ones,  and  are  those  in  terms  of  which 
the  problem  is  formulated.  Here  there  are  five  quantities  and  three 
fundamental  kinds  of  unit.  There  are  therefore  two  dimensionless 
products.  In  the  analysis  of  the  last  chapter  we  saw  that  in  finding 
the  dimensionless  products  we  had  to  solve  a  system  of  algebraic 
equations.  Certain  of  the  solutions  could  be  assigned  at  pleasure, 
and  the  others  determined  in  terms  of  them.  In  this  particular 
problem  we  are  interested  especially  in  t,  and  let  us  say  k.  Then  let 
us  choose  the  exponents  of  t  and  k  in  the  dimensionless  products 
as  those  which  are  to  be  assigned  at  pleasure  and  in  terms  of  which 
the  others  are  to  be  computed.  Now  the  algebraic  theorem  showed 
that  there  were  two  linearly  independent  sets  of  exponents  which 
we  might  assign  to  t  and  k,  and  that  it  is  possible  to  choose  these 
two  sets  in  an  infinite  number  of  ways.  We  will  try  to  select  the  two 
simplest.  For  the  present  purpose  we  will  accomplish  this  by  assign- 
ing the  value  1  to  the  exponent  of  t  and  0  to  that  of  k  for  the  one 
set,  and  0  to  the  exponent  of  t  and  1  to  the  exponent  of  k  for  the 
other  set.  This  is  certainly  a  simple  couple  of  pairs,  and  has  the 
effect  of  making  both  t  and  k  appear  in  only  one  dimensionless 
product.  "We  therefore  have  to  find  the  two  dimensionless  products 

t  vtti  d^1  g^i     ?     k  vaa  d^2  g?2. 

We  have  now  two  sets  of  algebraic  equations  for  the  two  sets  of 
unknown  exponents  a1?  ft,  yx  and  a2,  ft,  y2.  These  equations  are 


3a2-3ft+y2  +  0 
Oa2  +  0ft-y2-2 


EXAMPLES  OF  DIMENSIONAL  ANALYSIS  61 

The  solutions  are 


Hence  the  dimensionless  products  are 


and( 


and  the  solution  is 


-  -,. 


where  the  function  f  is  undetermined. 

Now  the  result  so  obtained  is  undoubtedly  correct  as  far  as  it 
goes,  but  an  examination  will  show  that  we  can  do  better,  and 
obtain  a  form  in  which  there  is  no  undetermined  function.  This 
improvement  can  be  effected  by  increasing  the  number  of  funda- 
mental units.  We  were  correct  in  using  the  ordinary  mechanical 
units,  for  the  equations  of  motion  involve  the  dynamical  relation 
between  force,  mass,  and  acceleration.  The  change  is  to  be  made  in 
a  direction  at  first  not  obvious  because  we  are  so  accustomed  to  using 
the  units  written  down.  It  is  evident  on  reflection,  however,  that  in 
the  equations  of  motion  governing  the  system  no  use  is  made  of  the 
fact  that  the  numerical  measure  of  the  volume  of  the  box  is  equal 
to  the  cube  of  the  length  of  one  of  its  linear  dimensions.  It  is  quite 
possible  to  measure  volumes  physically  in  terms  of  a  particular 
volume  chosen  as  unity  by  cutting  up  the  larger  volume  into  smaller 
volumes  congruent  with  the  unit,  and  counting  the  number  of 
times  that  the  unit  is  contained  in  the  larger  volume.  It  may  then 
be  proved  that  the  number  so  obtained  is  proportional  to  the  cube 
of  the  number  measuring  one  of  the  linear  dimensions.  In  fact,  this 
is  the  method  of  proof  originally  adopted  by  Euclid  in  dealing  with 
both  areas  and  volumes.  After  the  geometrical  fact  has  been  proved, 
it  becomes  natural  to  define  the  unit  volume  as  that  volume  which 
is  equal  to  a  cube  whose  sides  are  unity,  but  this  definition  and 
restriction  are  of  value  only  in  those  problems  in  which  the  relation 
between  volume  and  length  enter  essentially  into  the  result.  Such 
is  not  the  case  here,  because  the  volume  of  the  box  is  of  importance 
only  as  determining,  in  conjunction  with  the  density  of  the  liquid, 
the  mass  filling  the  box.  We  might  perfectly  well  measure  length 


62  DIMENSIONAL  ANALYSIS 

for  this  problem  in  inches,  and  the  volume  in  quarts,  provided,  of. 
course,  that  we  measure  density  as  mass  per  quart. 

Let  us  then  attempt  the  problem  again,  now  taking  volume  as 
an  independent  unit  of  its  own  kind.  Then  we  shall  have  : 

Name  of  Quantity.  Symbol.  Dimensional  Formula. 
Elastic  constant,  k  /  ^  MT~2 

Time  of  oscillation,  t  T 

Volume  of  box,  v  >/  ^ 

Density  of  liquid,  d  '  $  MV"1 

Acceleration  of  gravity,  g     &  ..—  LT~2 

We  have  now  five  variables,  but  four  fundamental  kinds  of  quan- 
tity, so  that  there  is  only  one  dimensionless  product.  We  are  par- 
ticularly interested  in  t,  so  we  choose  the  exponent  of  t  equal  to 
unity,  and  are  required  to  find  the  other  exponents  so  that 

t  ka  v£  d^  g6  is  dimensionless. 

This  problem  is  so  simple  that  we  can  solve  for  the  unknowns  by 
inspection,  or  if  we  prefer,  write  out  the  equations,  which  are  : 


8  =  0 

-2a-28  +  lrz:0 
£-7  =  0 

The  solution  of  this  set  of  equations  is 

«=%,js  =  -&y  =  -%,8  =  o. 

The  dimensionless  product  is 

tk*v-*d-*, 

and  the  solution  is 

t  =  Const  /?• 

The  information  embodied  in  this  solution  is  evidently  much 
greater  than  in  the  more  noncommittal  one  obtained  with  three 
units.  It  is  seen  from  the  new  solution,  for  example,  that  the  time 
of  oscillation  does  not  depend  on  the  intensity  of  gravity.  Physically, 
of  course,  this  means  that  gravity  is  effective  only  in  changing  the 
mean  position  of  equilibrium;  as  gravity  increases  the  weight  is 
pulled  down  and  oscillates  about  a  position  nearer  the  center  of 


EXAMPLES  OF  DIMENSIONAL  ANALYSIS  63 

attraction,  but  the  period  of  oscillation  is  not  changed  thereby.  It 
was  not  at  all  obvious  or  necessary  from  the  first  form  of  the  solu- 
tion that  the  time  would  be  independent  of  gravity,  but  that  the 
previous  solution  is  not  inconsistent  with  this  one  is  seen  by  putting 
the  f  of  the  previous  solution  equal  to  a  constant  times  the  inverse 
square  root  of  the  argument,  when  the  two  solutions  become 
identical. 

Instead  of  increasing  the  number  of  fundamental  units  from 
three  to  four,  we  might  have  obtained  the  same  result  by  observing 
that  the  equations  of  motion  are  concerned  only  with  the  total  mass 
on  the  end  of  the  string,  and  hence  the  volume  and  the  density  can 
affect  the  result  only  in  so  far  as  they  enter  through  their  product, 
the  mass.  According  to  this  method  of  treatment  we  would  have  put 
v  and  d  together  as  one  quantity,  so  that  we  would  have  been  con- 
cerned with  only  four  quantities  and  three  fundamental  kinds  of 
unit.  The  result  would  have  been  the  same  as  by  the  method  which 
we  adopted.  In  fact,  it  will  often  be  found  possible  by  using  special 
knowledge  of  the  problem  to  obtain  in  this  way  more  detailed  infor- 
mation than  would  have  been  possible  by  the  general  analysis. 

If  we  use  the  mass  as  one  of  the  variables,  the  result  assumes  the 

form  t  =  Const  V f ,  and  again  we  have  a  dimensionless  product  of 
fewer  than  the  normal  number  of  terms. 

Now  let  us  consider  a  problem  illustrating  how  it  is  that  the 
result  is  unaffected  by  increasing  the  number  of  units  if  at  the  same 
time  the  number  of  dimensional  constants  is  increased.  We  .take  the 
same  problem  as  above,  except  that  we  now  give  only  the  mass  on 
the  end  of  the  spring,  and  do  not  attempt  to  analyze  the  mass  into 
volume  times  density.  The  variables  will  be  mass  (m),  time  of  oscil- 
lation (t),  and  stiffness  of  spring  (k).  "We  can  omit  the  intensity  of 
gravity,  because  we  have  already  seen  it  to  be  without  effect.  In 
discussing  this  problem  we  propose  to  use  five  fundamental  kinds  of 
unit,  which  we  will  choose  as  mass,  length,  time,  as  usual,  and  in 
addition  force,  and  velocity.  This  problem  is  evidently  one  in 
mechanics  involving  in  the  statement  of  the  relations  between  the 
parts  the  experimental  fact  that  force  is  proportional  to  mass  times 
accleration.  Hence  in  formulating  the  equations  of  motion  we  will 
have  to  introduce  this  proportionality  factor,  which  will  appear  in 
the  analysis  as  a  new  dimensional  constant.  This  factor  is  to  connect 
force,  mass,  and  acceleration.  But  now  acceleration  must  be  rede- 


64  DIMENSIONAL  ANALYSIS 

fined  if  we  are  using  velocity  as  a  unit  of  its  own  kind.  Acceleration 
will  now  be  denned  as  time  rate  of  change  of  velocity,  and  will  have 
the  dimensions  VT™1.  The  equation  of  motion  thus  written  will 
express  a  relation  between  the  force  and  the  velocity  and  the  time. 
But  the  force  is  connected  with  the  displacement  through  the 
elastic  constant,  so  that  to  solve  the  equations  a  relation  is  needed 
between  displacement,  velocity,  and  time.  The  experimental  fact,  of 
course,  is  that  velocity  is  proportional  to  the  quotient  of  distance 
by  time.  The  factor  of  proportionality  will  appear  in  the  final  result 
as  a  dimensional  constant.  We  now  have  our  list  of  quantities  com- 
plete. They  compose  three  physical  variables,  and  two  dimensional 
constants. 

Name  of  Quantity.  Symbol.  Dimensional  Formula. 

Time  of  oscillation,  t  T 

Mass  at  end  of  spring,  m  M 

Stiffness  of  spring,  k  FL"1 

The  force  dimensional  constant,  f  FM"1  TV"1 
The  velocity  dimensional  con- 

stant, v  L-1  TV 

Here  F  is  the  dimensional  symbol  of  force  measured  in  units  of 
force,  and  V  the  dimensional  symbol  of  velocity.  The  dimensional 
formulas  were  obtained  by  the  regular  methods,  noting  only  that 
the  stiffness  of  the  spring  is  defined  as  the  force  exerted  by  the 
spring  per  unit  displacement  of  the  end. 

We  have  now  to  find  the  dimensionless  products  involving  these 
five  variables.  We  note  in  the  first  place  that  there  are  five  variables, 
and  five  fundamental  kinds  of  quantity,  so  that  in  general  there 
would  be  no  dimensionless  product.  But  it  may  be  seen  on  writing 
it  out  that  the  determinant  of  the  exponents  in  the  dimensional 
formulas  vanishes,  so  that  in  this  special  case  there  is  a  dimension- 
less  product  with  fewer  than  the  normal  number  of  factors.  Of 
course  we  knew  that  this  must  be  the  case  from  our  previous  discus- 
sion. Now,  as  before,  we  select  t  as  the  quantity  in  which  we  are 
particularly  interested,  write  the  dimensionless  product  in  the  form 


and  write  down  the  condition  that  the  product  is  dimensionless.  This 
gives 


EXAMPLES  OF  DIMENSIONAL  ANALYSIS  65 

a  —  7  =  0  condition  on  M 

—  ft  —  8  =  0  condition  on  L 
7  +  8  +  1  =  0  condition  on  T 

(3  +  y  =  0  condition  on  F 

—  y  +  8  =  0  condition  on  V. 

The  solution  is 

a  =  -y2,  p  =  y2,y  =  ~y2,  <  =  -y2. 

The  dimensionless  product  is 

tm-i  k*f-*  v-*, 
and  the  final  solution 

t  =  Const  V^-' 

This  is  exactly  the  same  as  the  solution  already  obtained,  on  put- 
ting the  dimensional  constants  f  and  v  equal  to  unity,  which  of 
course  was  their  value  in  the  ordinary  mechanical  system  of  units. 

Although  this  example  gives  no  new  results,  it  is  instructive  in 
showing  that  any  system  of  fundamental  units  whatever  is  allow- 
able, provided  only  that  the  dimensional  constants  required  by  the 
special  problem  are  also  introduced. 

We  now  consider  a  problem  in  which  it  is  an  advantage  to  treat 
force  as  a  unit  of  its  own  kind.  This  is  the  problem  of  Stokes  of  a 
small  sphere  falling  under  gravity  in  a  viscous  liquid.  The  sphere 
is  so  small  that  the  motion  is  everywhere  slow,  so  that  there  is 
nowhere  turbulence  in  the  fluid.  The  elements  with  which  we  have 
to  deal  in  this  problem  are  the  velocity  of  fall;  the  density  of  the 
sphere,  the  diameter  of  the  sphere,  the  density  of  the  liquid,  the 
viscosity  of  the  liquid,  and  the  intensity  of  gravity.  The  problem  is 
evidently  one  in  mechanics,  so  that  if  we  use  the  ordinary  mechani- 
cal units  there  will  be  no  dimensional  constants  to  introduce.  But 
we  notice  that  the  problem  is  of  a  very  special  kind  for  a  mechanical 
problem.  The  motion  is  slow,  and  the  velocity  is  steady,  the  forces 
acting  on  the  sphere  and  the  liquid  being  everywhere  held  in  equi- 
librium by  the  forces  called  out  by  the  viscosity  of  the  liquid. 
That  is,  although  this  is  a  problem  involving  motion,  it  is  a  problem 
involving  unaccelerated  motion,  and  the  forces  are  in  equilibrium 
everywhere.  The  problem  is  essentially,  therefore,  one  in  statics,  and 
in  solving  the  problem  we  need  to  make  no  use  of  the  fact  that  in 
those  cases  where  there  happens  to  be  an  acceleration  the  force  is 


66  DIMENSIONAL  ANALYSIS 

proportional  to  mass  times  acceleration.  In  this  problem,  therefore, 
we  treat  force  as  its  own  kind  of  quantity,  and  do  not  have  to  intro- 
duce a  compensating  dimensional  constant.  Our  analysis  of  the 
problem  is  now  as  follows  : 

Name  of  Quantity.  Symbol.            Dimensional  Formula. 

Velocity  of  fall,  v  LT-1 

Diameter  of  sphere,  D  L 

Density  of  sphere,  dt  ML~3 

Density  of  liquid,  d2  ML~3 

Viscosity  of  liquid,  /*  FL~2T 

Intensity  of  gravity,  g  FM"1 

The  dimensional  formula  of  viscosity  is  obtained  directly  from 
its  definition  as  force  per  unit  area  per  unit  velocity  gradient.  The 
intensity  of  gravity  is  taken  with  the  dimensions  shown,  because 
obviously  the  equations  of  motion  will  not  mention  the  accelerational 
aspect  of  gravitational  action,  but  only  the  intensity  of  the  force 
exerted  by  gravity  upon  unit  mass. 

We  now  have  six  variables,  and  four  fundamental  kinds  of  unit. 
There  are,  therefore,  two  dimensionless  products.  One  of  them  is 
evident  on  inspection,  and  is  d2/d±.  Now  of  the  remaining  quantities 
we  are  especially  interested  in  v.  We  need  combine  this  with  only 
four  other  quantities  to  obtain  a  dimensionless  product.  We  choose 
D,  di,  JM,  and  g,  and  seek  a  dimensionless  product  of  the  form 


The  exponents  are  at  once  found  to  be 

«=-2,/T:=  -1/8==  -1,^  =  1 

Hence  the  dimensionless  products  are 

v  D-2  dr1  n  g-1  and 
and  the  final  solution  is 


The  function  f  is  arbitrary,  so  that  we  cannot  tell  how  the  result 
depends  on  the  densities  of  the  sphere  and  the  liquid,  but  we  do  see 
that  the  velocity  of  fall  varies  as  the  square  of  the  diameter  of  the 


EXAMPLES  OF  DIMENSIONAL  ANALYSIS  67 

sphere,  and  the  intensity  of  gravity,  and  inversely  as  the  viscosity 
of  the  fluid. 

This  problem  has  of  course  long  been  solved  by  the  methods  of 
hydrodynamics,  and  the  solution  is 


See,  for  example,  Millikan,  Phys.  Rev.,  2,  110,  1913.  The  exact  solu- 
tion  is  obtained   from  the   more   general   one   above   by   giving 

)  as  the  special  value  of  the  function. 


If  we  had  solved  this  problem  with  the  ordinary  mechanical  units, 
in  which  force  is  defined  as  mass  times  acceleration,  we  should  have 
had  'three  instead  of  two  dimensionless  products,  and  the  final  result 
would  have  been  of  the  form 


d.D 


In  this  form  we  evidently  can  say  nothing  about  the  effect  on  the 
velocity  of  any  of  the  elements  taken  by  themselves,  since  they  all 
occur  under  the  arbitrary  functional  symbol. 

There  are  many  problems  in  which  some  specific  information 
about  the  nature  of  the  physical  system  enables  the  information 
given  by  dimensional  analysis  to  be  supplemented  so  that  a  more 
restricted  form  of  the  solution  can  be  obtained  than  would  be  pos- 
sible by  dimensional  analysis  alone.  There  is,  of  course,  no  law 
against  combining  dimensional  analysis  with  any  information  at  our 
command. 

Let  us  take  as  a  simple  example  the  discussion  of  the  problem  of 
the  bending  of  a  beam.  This  is  a  problem  in  elasticity.  Let  us  en- 
deavor to  find  how  the  stiffness  of  the  beam  depends  on  the  dimen- 
sions of  the  beam,  and  any  other  quantities  that  may  be  involved. 
Now  the  equations  of  elasticity  are  equations  of  ordinary  mechanics. 
The  mechanical  system  of  three  fundamental  units  is  indicated. 
The  equations  of  elasticity  from  which  the  solution  is  to  be  obtained 
will  involve  the  elastic  constants  of  the  material.  If  the  material  is 
isotropic,  there  will  be  two  elastic  constants,  which  may  be  chosen  as 
Young's  modulus,  and  the  shear  modulus.  Out  analysis  may  now 
run  as  follows: 


68  DIMENSIONAL  ANALYSIS 

Name  of  Quantity.  Symbol.            Dimensional  Formula. 

Stiffness  (Force/deflection),         S  MT~2 

Length,  1  L 

Breadth,  b  L 

Depth,  d  L 

Young's  modulus,  E  ML"1  T~2 

Shear  modulus,  ju  ML-1  T~2 

There  are  six  variables,  and  three  fundamental  kinds  of  unit. 
Hence  according  to  the  general  rule  there  should  be  three  dimen- 
sionless  products.  Three  such  products  can  obviously  be  written 
down  by  inspection,  and  are 

b/1,  d/1,  and  /x/E. 

Now  none  of  these  dimensionless  products  contains  the  quantity  S 
in  which  we  are  particularly  interested,  and  it  is  evident  that  there 
is  something  peculiar  about  this  problem.  It  will  in  fact  be  found, 
on  going  back  to  the  system  of  algebraic  equations  on  which  the 
solution  depends,  and  writing  down  the  matrix  of  the  coefficients 
obtained  from  the  exponents  in  the  dimensional  formulas,  that  each 
of  the  three  rowed  determinants  formed  out  of  the  matrix  is  zero. 
This  is  evident  on  inspection  of  the  matrix. 

100011 
0        1        1        l_l_l 
-2        0        0        0-2-2 

This  means  that  in  this  particular  case  there  are  more  dimensionless 
products  than  are  given  by  the  general  rule.  That  such  is  the  case 
should  have  been  evident  beforehand.  In  the  first  place,  an  inspec- 
tion of  the  dimensional  formulas  shows  that  M  and  T  always  enter 
in  the  combination  MT~2,  so  that  this  combination  together  might 
have  been  treated  as  a  fundamental  unit  itself,  so  that  there  would 
have  been  only  two  fundamental  units  instead  of  three,  and  four 
instead  of  three  dimensionless  products.  In  the  second  place,  this  is 
a  problem  in  statics,  in  which  mass  and  time  do  not  enter  into  the 
results.  The  dimensions  of  all  the  quantities  could  have  been  given 
in  terms  of  force  and  length  as  the  fundamental  units.  This  remark 
is  the  physical  equivalent  of  the  analytical  observation  that  M  and 
T  always  occur  in  the  combination  MT~2  (force  is  MT~2  multiplied 
byL). 


EXAMPLES  OF  DIMENSIONAL  ANALYSIS  69 

With  the  knowledge  that  there  is  still  another  dimensionless 
product,  we  can  see  by  inspection  that  it  is 

S/El, 
so  that  the  final  solution  is 


b  d   /A 

7  ?  i; 


=  Blf 


This  solution  gives  no  information  about  the  variation  of  stiffness 
with  the  dimensions  of  the  beam.  Now  it  is  obvious  from  elementary 
considerations  of  elasticity,  that  for  slender  beams,  the  stiffness 
must  be  approximately  proportional  to  the  breadth,  other  things 
being  equal,  for  the  boundary  conditions  are  such  that  the  solution 
for  a  beam  of  twice  the  breadth  may  be  obtained  approximately  by 
simply  placing  beside  each  other  two  of  the  original  beams.  Hence 

f  must  be  of  such  a  form  that  If  (-,-,  —  )  reduces  to  b  <£  (  -,  —  ) 

\1  1  E/  \1  B/ 

and  the  value  of  f  must  obviously  be  -  <£[-,—  ).  The  restricted 

1      \l  E/ 

solution  is  therefore 


I'E 

The  solution  now  shows  that  a  beam  of  double  the  length  can  be 
kept  of  the  same  stiffness  by  doubling  the  depth.  The  detailed  solu- 
tion of  elasticity  shows  that  the  ratio  of  d  to  1  enters  as  the  cube,  as  a 
factor  of  proportionality,  so  that  the  stiffness  is  proportional  directly 
to  the  cube  of  the  depth,  inversely  to  the  cube  of  the  length,  directly 
as  the  breadth,  and  to  some  unknown  function  of  the  elastic 
constants. 

This  method  of  supplementing  the  results  of  dimensional  analysis 
by  other  information  will  often  be  found  of  value.  There  are  numer- 
ous examples  in  Lord  Rayleigh's  treatments.  Rayleigh  does  not 
always  separate  the  analysis  into  a  dimensional  and  another  part, 
but  states  that  a  result  can  be  proved  by  dimensional  analysis, 
although  it  may  require  supplementing  in  some  such  way  as  above. 
A  good  example  will  be  found  in  his  treatment  of  the  scattering  of 
light  by  the  sky.2  The  result  that  the  scattering  varies  inversely  as 
the  fourth  power  of  the  wave  length  of  the  incident  light  is  obtained 
by  using  in  addition  to  dimensional  knowledge  the  fact  that  "From 


W  • 

70  DIMENSIONAL  ANALYSIS 

what  we  know  of  the  dynamics  of  the  situation  i  (ratio  of  ampli- 
tude of  incident  and  scattered  light)  varies  directly  as  T  (volume  ' 
of  scattering  particle)   and  inversely  as  r   (distance  of  point  of 
observation  from  scattering  particle)." 

Thus  far  we  have  considered  only  problems  in  mechanics,  but  of 
course  the  method  is  not  restricted  to  such  problems,  but  can  be 
applied  to  any  system  whose  laws  can  be  formulated  in  a  form 
independent  of  the  size  of  the  fundamental  units  of  measurement. 

Let  us  consider,  for  example,  a  problem  from  the  kinetic  theory 
of  gases,  and  find  the  pressure  exerted  by  a  perfect  gas.  The  atoms 
of  the  gas  in  kinetic  theory  are  considered  as  perfect  spheres,  com- 
pletely elastic,  and  of  negligible  dimensions  compared  with  their 
distance  apart.  The  only  constant  with  dimensions  required  in 
determining  the  behavior  of  each  atom  is  therefore  its  mass.  The 
behavior  of  the  aggregate  of  atoms  is  also  evidently  characterized 
by  the  density  of  the  gas  or  the  number  of  atoms  per  unit  volume. 
The  problem  is  evidently  one  of  mechanics,  and  the  pressure  exerted 
by  the  gas  is  to  be  found  by  computing  the  change  of  momentum 
per  unit  time  and  per  unit  area  of  the  atoms  striking  the  walls  of 
the  enclosure.  The  mechanical  system  of  units  is  therefore  indicated. 
But  in  addition  to  the  ordinary  mechanical  features  there  is  the 
element  of  temperature  to  be  considered.  How  does  temperature 
enter  in  writing  down  the  equations  of  motion  of  the  system? 
Obviously  through  the  gas  constant,  which  gives  the  average  kinetic 
energy  of  each  atom  as  a  function  of  the  temperature.  Our  analysis 
of  the  problem  therefore  runs  as  follows : 

Name  of  Quantity.  Symbol.            Dimensional  Formula. 

Pressure  exerted  by  gas,  p  ML"1  T~2 

Mass  of  the  atom,  m  M 

Number  of  atoms  per  unit  of 

volume,  N  L~8 

Absolute  temperature,  6  0 

Gas  constant  per  atom,  k  ML2  T~2  0"1 

"We  have  here  five  variables,  and  four  kinds  of  units.  There  is, 
therefore,  one  dimensionless  product.  Since  p  is  the  quantity  in 
which  we  are  interested,  we  choose  the  exponent  of  this  as  unity. 
We  have  to  find 

p  m« 


EXAMPLES  OF  DIMENSIONAL  ANALYSIS  71 

The  work  of  solution  is  as  before.  The  values  of  the  exponents  are 

a  =  0,  £=-1,  y  =  -l,  8  =  -l. 
The  dimensionless  product  is 

p  N-1  0-1  k-1, 
and  the  final  solution  is 

p  =  Const  N  k  0. 

That  is,  the  pressure  is  proportional  to  the  gas  constant,  to  the 
density  of  the  gas,  and  to  the  absolute  temperature,  and  does  not 
depend  on  the  mass  of  the  individual  atoms.  The  formula  for  pres- 
sure is,  of  course,  one  of  the  first  obtained  in  any  discussion  of 
kinetic  theory,  and  differs  from  the  above  only  in  that  the  numerical 
value  of  the  constant  of  proportionality  is  determined. 

In  this  problem,  or  in  other  problems  of  the  same  type,  we  could, 
if  we  preferred,  eliminate  temperature  as  an  independent  kind  of 
variable  and  define  it  as  equal  to  the  energy  of  the  atom.  This 
amounts  merely  to  changing  the  size  of  the  degree,  but  does  not 
change  the  ratio  of  any  two  temperatures,  and  is  the  sort  of  change 
of  unit  which  is  required  according  to  the  fundamental  assumptions. 
If  we  define  temperature  in  this  way,  the  gas  constant  is  of  course 
to  be  put  equal  to  unity.  We  would  now  have  three  fundamental 
units,  and  four  variables.  There  is  of  course  again  only  one  dimen- 
sionless product,  and  the  same  result  would  be  obtained  as  before. 
Let  us  go  through  the  work ;  it  is  instructive. 

Name  of  Quantity.  Symbol.  Dimensional  Formula. 

Pressure  exerted  by  gas,  p  ML"1  T~2 

Mass  of  atom,  m  M 

Number  of  atoms  per  unit  vol- 
ume, N  L~3 
Absolute  temperature,                    0                           ML2  T~2 

We  now  have  to  find  our  dimensionless  product  in  the  form 

p  ma  N£  Ov. 
The  exponents  are  at  once  found  to  be 

a  =  0,£=-]L,r  =  --l, 
and  the  solution  of  the  problem  is 

p  —  Const  N  6. 


72  DIMENSIONAL  ANALYSIS 

This  solution  is  like  the  one  obtained  previously  except  for  the 
presence  of  the  gas  constant,  but  since  the  gas  constant  in  the  new 
system  of  units  is  unity,  the  two  solutions  are  identical,  as  they 
should  be. 

This  procedure  can  obviously  be  followed  in  any  problem  whose 
solution  involves  the  gas  constant.  Temperature  may  be  either 
chosen  as  an  independent  unit,  in  which  case  the  gas  constant 
appears  explicitly  as  a  variable,  or  temperature  may  be  so  denned 
that  the  gas  constant  is  always  unity,  and  temperature  has  the 
dimensions  of  energy.  The  same  procedure  is  not  incorrect  in  prob- 
lems not  involving  the  gas  constant  in  the  solution.  But  if  in  this 
class  of  problem  temperature  is  denned  as  equal  to  the  kinetic 
energy  of  an  atom  (or  more  generally  equal  to  the  energy  of  a 
degree  of  freedom)  and  the  gas  constant  is  made  equal  to  unity, 
the  fundamental  units  are  restricted  with  no  compensating  advan- 
tage, so  that  although  the  results  are  correct  as  far  as  temperature 
is  proportional  to  the  energy  of  a  degree  of  freedom,  they  will  not 
give  so  much  information  as  might  have  been  obtained  by  leaving 
the  units  less  restricted. 

It  is  obvious  that  these  remarks  apply  immediately  to  the  heat 
transfer  problem  of  Eayleigh  treated  in  the  introductory  chapter. 

Many  persons  feel  an  intuitive  uncertainty  with  regard  to  the 
dimensions  to  be  assigned  to  temperature.  This  is  perhaps  because 
of  the  feeling  that  a  dimensional  formula  is  a  statement  of  the 
physical  nature  of  the  quantity  as  contained  in  the  definition.  Now 
the  absolute  temperature,  as  we  have  used  it  above,  is  the  thermo- 
dynamic  absolute  temperature,  defined  with  relation  to  the  second 
law  of  thermodynamics.  It  is  difficult  to  see  how  such  a  complex 
of  physical  operations  as  is  involved  in  the  use  of  the  second  law 
(such  as  Kelvin  first  gave  in  his  definition  of  absolute  temperature) 
can  be  reproduced  in  a  simple  dimensional  formula.  It  is,  however, 
evident  that  measurements  of  energy,  for  example,  are  involved  in 
an  application  of  the  second  law,  so  that  perhaps  in  some  way  the 
ordinary  mechanical  units  ought  to  be  involved  in  the  dimensional 
formula.  But  we  have  seen  that  the  dimensional  formula  is  con- 
cerned only  with  an  exceedingly  restricted  aspect  of  the  way  in 
which  the  various  physical  operations  enter  the  definition,  namely 
with  the  way  in  which  the  numerical  measure  of  a  quantity  changes 
when  the  fundamental  units  change  in  magnitude.  Now  a  little 
reflection  shows  that  any  such  procedure  as  that  of  Lord  Kelvin 


EXAMPLES  OF  DIMENSIONAL  ANALYSIS  73 

applied  to  the  definition  of  absolute  temperature  through  the  use 
of  the  second  law  imposes  on  the  number  measuring  a  given  con- 
crete temperature  no  restriction  whatever  in  terms  of  the  units  in 
which  heat  or  energy,  for  example,  are  measured.  The  size  of  a 
degree  of  thermodynamic  temperature  may  be  fixed  entirely  arbi- 
trarily so  that  there  are  any  number  of  degrees  between  the  freez- 
ing and  the  boiling  points  of  water,  for  example,  absolutely  without 
reference  to  the  size  of  any  other  unit.  We  are  concerned  in  the 
dimensional  formula  with  the  definition  in  terms  of  the  second  law 
only  in  so  far  as  this  definition  satisfies  the  principle  of  the  absolute 
significance  of  relative  magnitude,  that  is,  the  principle  that  the 
ratio  of  the  measures  of  two  concrete  examples  shall  be  independent 
of  the  size  of  the  units.  Now  it  is  evident  that  the  thermodynamic 
definition  of  absolute  temperature  does  leave  the  ratio  of  any  two 
concrete  temperatures  independent  of  the  size  of  the  units.  The 
dimensional  formula  of  temperature,  therefore,  need  contain  no 
other  element*  and  temperature  may  be  treated  as  having  its  own 
dimensions. 

There  is  no  necessity  in  using  the  absolute  thermodynamic  tem- 
perature. We  might,  for  instance,  define  the  number  of  degrees  in  a 
given  temperature  interval  as  the  number  of  units  of  length  which 
the  kerosene  in  a  certain  capillary  projecting  from  a  certain  bottle 
of  kerosene  moves  when  the  bottle  is  brought  from  one  temperature 
to  another.  The  temperature  so  defined  evidently  satisfies  the  prin- 
ciple of  the  absolute  significance  of  relative  magnitude,  for  if  the 
size  of  the  unit  of  length  measured  along  the  capillary  is  cut  in  half, 
the  number  of  degrees  in  every  temperature  interval  is  doubled. 
The  advantage  of  the  thermodynamic  scale  is  one  of  simplicity;  in 
the  kerosene  scale  the  behavior  of  a  perfect  gas  could  not  be  charac- 
terized in  terms  of  a  single  constant,  and  the  Fourier  equations  of 
heat  conduction  could  not  be  written,  except  over  a  very  narrow 
range,  in  terms  of  a  single  coefficient  of  thermal  conductivity. 

Besides  the  question  of  the  dimensions  of  temperature,  there  is 
one  other  question  connected  with  the  application  of  dimensional 
analysis  to  problems  in  thermodynamics  which  is  apt  to  be  puzzling ; 
this  is  the  matter  of  the  so-called  logarithmic  constants.  In  books  on 
thermodynamics  equations  are  very  common  which  on  first  sight 
do  not  appear  to  be  complete  equations  or  to  be  dimensionally 
homogeneous.  These  equations  often  involve  constants  which  cannot 
change  in  numerical  magnitude  by  some  factor  when  the  size  of  the 


74  DIMENSIONAL  ANALYSIS 

fundamental  units  is  changed,  but  must  change  by  the  addition  of 
some  term.  An  example  will  be  found  on  page  6  of  Nernst's  Yale 
lectures  on  the  Applications  of  Thermodynamics  to  Chemistry.  This 
equation  is  : 

logC  =  - 


RT      R  R         2R 

In  this  equation  C  is  a  concentration  of  a  given  gaseous  sub- 
stance, A0  is  a  heat,  a,  b,  and  c  are  dimensional  coefficients  in  the 
usual  sense  into  which  we  need  not  inquire  further,  except  that  a/R 
is  dimensionless,  and  i  is  a  constant  of  integration.  It  is  obvious  that 
this  formula  as  it  stands  does  not  allow  the  size  of  the  fundamental 
units  to  be  changed  by  making  the  usual  sort  of  changes  in  the 
various  quantities.  But  a  rearrangement  of  terms  is  possible  which 
throws  the  formula  into  the  conventional  form.  If  we  group  together 
the  terms  log  C,  a/R  log  T,  and  i  into  the  single  term 

C 
log—— 


where  i'  is  a  new  constant,  we  evidently  have  a  complete  equation  in 


a 


the  usual  sense  of  the  word,  and  i'  has  the  dimensions  of  C  T    R. 

This  sort  of  rearrangement  of  terms  is  always  possible  if  the 
formula  has  had  a  theoretical  derivation,  as  have  all  the  formulas 
of  these  treatises,  and  the  logarithmic  constant  appears  only  as  a 
formal  exception. 

The  logarithmic  constant  is  met  with  so  often  in  thermodynamic 
formulas  because  in  most  thermodynamic  expressions  there  is  an 
undetermined  constant  of  integration  arising  from  the  fact  that 
energy,  or  work,  or  entropy,  or  thermodynamic  potential  has  no 
absolute  significance,  but  is  only  the  difference  between  two  values, 
and  the  coordinates  of  the  initial  point  which  fix  the  origin  of 
entropy,  for  example,  may  be  chosen  at  pleasure. 

The  formulas  of  thermodynamics  also  often  present  a  strange 
appearance  in  the  way  that  concrete  quantities  (that  is,  quantities 
with  dimensions)  appear  as  the  arguments  of  transcendental  func- 
tions. Thus  on  page  5  of  the  same  book  of  Nernst's,  we  find  the 
formula. 


dT 


EXAMPLES  OF  DIMENSIONAL  ANALYSIS  75 

This  comes  from  an  application  of  Clapeyron's  equation  to  a  sub- 
stance whose  vapor  obeys  the  perfect  gas  law,  and  the  volume  of 
whose  vapor  is  large  compared  with  that  of  the  liquid.  In  spite  of 
the  appearance  of  a  pressure  under  the  logarithm  sign,  this  equa- 
tion is  seen  on  examination  to  be  a  complete  equation,  and  holds 
valid  for  all  sizes  of  the  fundamental  units.  This  may  at  once  be 

seen  on  expanding  -  —  ,  which  is  equal  to  --  —  ,  and  is  therefore 
dT  pdT 

of  zero  dimensions  in  p.  Expressions  of  this  sort  in  which  the  loga- 
rithm is  taken  of  a  quantity  with  dimensions  are  particularly  com- 
mon in  thermodynamics,  and  often  arise  from  the  equations  of  the 
perfect  gas.  The  occurrence  of  such  logarithmic  terms  should,  it 
seems  to  me,  be  difficult  for  those  to  interpret  who  like  to  regard  a 
dimensional  formula  as  expressing  a  concrete  physical  operation  on 
a  concrete  physical  thing. 

That  the  occurrence  of  such  expressions  is  not  contrary  to  the  n 

theorem  is  seen  from  the  expanded  form,  --  —  .  The  slope  of  the 

p  dT 

curve,  —  —  ,  would  be  one  of  the  variables  in  which  the  dimensionless 

products  are  to  be  expressed,  and  there  is  evidently  no  exception. 
Our  theorem  has  merely  stated  that  the  results  are  expressible  in 
terms  of  dimensionless  products;  we  have  no  reason  to  think  that 
the  man  who  derived  the  formula  was  accommodating  enough  to 
write  the  formula  so  that  this  would  appear  without  some  rearrange- 
ment of  the  terms. 

Let  us  close  this  chapter  of  special  examples  with  several  electri- 
cal examples. 

As  the  first  example  consider  an  electric  circuit  possessing  capa- 
city and  inductance.  An  oscillatory  discharge  is  excited  in  it.  How 
does  the  period  of  the  discharge  depend  on  the  constants  of  the 
circuit?  The  solution  of  this  problem  is  to  be  obtained  from  the 
detailed  equations  of  the  electric  circuit,  written  in  the  usual  form, 
in  electromagnetic  units.  None  of  the  electrostatic  effects  of  the  cur- 
rent, or  the  interactions  with  a  magnet,  have  to  be  considered  in 
the  equations,  which  are  of  the  form 


d  t 


76  DIMENSIONAL  ANALYSIS 

Hence  in  establishing  the  units  fundamental  to  this  equation  it  is 
evidently  sufficient  to  consider  only  three  fundamental  kinds  of 
quantity,  namely,  quantity  of  electricity,  time,  and  energy.  Then 
current  is  to  be  defined  as  quantity  per  unit  time,  coefficient  of  self- 
induction  is  such  a  quantity  that  when  multiplied  by  half  the  square 
of  the  current  it  gives  an  energy,  and  similarly  the  capacity  is  such 
a  quantity  that  it  gives  an  energy  when  divided  into  the  square  of  a 
quantity.  We  may  now  make  our  usual  analysis  of  the  problem. 

Name  of  Quantity.  Symbol.  Dimensional  Formula. 
Quantity  of  electricity,  q  Q 

Current,  i  QT"1 

Coefficient  of  self-induction,          L  Q~2T2E 

Capacity,  c  Q^"1 

Periodic  time,  t  T 

Now  the  time  of  oscillation  might  conceivably  involve  the  con- 
stants of  the  circuit,  and  the  initial  charge  of  the  condenser.  That 
is,  we  are  to  look  for  a  relation  between  q,  L,  c,  and  t.  Since  we  are 
especially  interested  in  t,  we  try  to  find  a  dimensionless  product  of 
the  form 

t  L*  c^  i?. 

The  exponents  are  at  once  found  to  be 


giving  as  the  solution  of  the  problem 


t  =  Const     ^  c. 


This  of  course  is  the  solution  which  would  be  found  by  actually 
solving  the  equations  of  the  circuit,  except  for  the  value  of  the  con- 
stant coefficient.  It  is  to  be  noticed  that  the  initial  charge  does  not 
enter.  This  problem  is  evidently  the  electrical  analogue  of  the 
mechanical  problem  of  the  simple  pendulum. 

It  is  perhaps  worth  noticing  again  that  some  knowledge  of  the 
nature  of  the  solution  is  necessary  before  dimensional  analysis  can 
be  used  to  advantage.  The  Australian  bushman,  when  attacking  this 
problem  for  the  first  time,  might  be  tempted  to  look  for  a  relation 
of  the  dimensions  of  a  time  between  the  constants  of  the  circuit, 
and  the  instantaneous  current,  and  the  instantaneous  charge  in  the 
condenser.  If  he  had  included  i  in  his  list  of  variables,  he  would  have 


EXAMPLES  OF  DIMENSIONAL  ANALYSIS  77 

found  that  q/i  also  has  the  dimensions  of  a  time,  and  his  solution 
would  have  been  of  the  form 

L 


t  =  /L  c  f  ( 

\ 


which  is  not  incorrect,  since  it  reduces  to  the  form  already  found  on 
putting  f  equal  to  a  constant,  but  it  gives  less  information  than  the 
previous  form. 

"We  now  consider  a  problem  in  electrostatics.  The  conception  of 
the  medium  introduced  by  Faraday  tells  us  that  it  is  possible  to 
regard  the  medium  as  the  seat  of  the  essential  phenomena  in  the 
electrostatic  field,  and  that  the  condition  of  the  medium  at  any 
instant  is  uniquely  determined  by  the  electric  vector  at  that  point. 
Let  us  seek  for  the  connection  between  the  space  density  of  energy 
in  the  electrostatic  field  and  the  intensity  of  the  field.  Since  this  is  a 
problem  in  statics,  the  phenomena  can  be  adequately  described  in 
terms  of  two  fundamental  units,  those  of  force  and  length.  Further- 
more the  field  equations  of  electrostatics  contain  no  dimensional  con- 
stants, so  that  the  velocity  of  light  does  not  enter  the  results,  as  it 
did  the  problem  of  the  mass  of  the  spherical  distribution  of  charge. 
In  terms  of  the  two  fundamental  units  of  force  and  length  we  may 
make  our  fundamental  definitions  as  follows.  Unit  electrostatic 
charge  is  that  charge  which  at  distance  of  unity  from  an  equal 
charge  in  empty  space  exerts  on  it  a  force  of  unity.  The  electric 
vector  is  that  vector  which  when  multiplied  into  the  charge  gives  the 
force  on  the  charge.  The  dielectric  constant  is  the  ratio  of  the  force 
between  two  charges  when  in  empty  space,  and  when  surrounded 
by  the  medium  in  question.  The  dimensions  of  dielectric  constant 
are  obviously  zero.  The  dimensions  of  energy  with  this  system  of 
units  are  obviously  force  multiplied  by  distance. 

We  now  formulate  the  problem. 

Name  of  Quantity.  Symbol.  Dimensional  Formula. 

Charge,  e 

Field  strength,  E 

Energy  density,  u  FL~2 

We  are  to  seek  for  a  relation  between  E  and  u.  Generally  there 
would  not  be  a  relation  between  these  quantities,  because  there  are 
two  fundamental  quantities  and  two  variables.  But  under  the  spe- 


78  DIMENSIONAL  ANALYSIS 

cial  conditions  of  this  problem  a  relation  exists,  and  the  result  is 
obvious  on  inspection  to  be 

u  =  Const  E2. 

In  treatises  on  electrostatics  the  constant  is  found  to  be  %. 

If  instead  of  the  energy  density  in  empty  space  we  had  tried  to 
find  the  energy  density  inside  a  ponderable  body  with  dielectric 
constant  e,  the  above  result  would  have  been  modified  by  the  appear- 
ance of  an  arbitrary  function  of  the  dielectric  constant  as  a  factor. 
Dimensional  analysis  can  give  no  information  as  to  the  form  of  the 
function.  As  a  matter  of  fact,  the  function  is  equal  to  the  dielectric 
constant  itself. 

This  problem  is  instructive  in  showing  the  variety  of  ways  in 
which  it  is  possible  to  choose  the  fundamental  units.  Since  the  prob- 
lem is  one  which  may  be  reduced  to  formulation  in  mechanical 
terms  (the  definitions  of  electrical  quantities  are  given  immediately 
in  terms  of  mechanical  quantities)  we  might  have  used  the  ordinary 
three  units  of  mechanics  as  fundamental,  and  written  the  dimen- 
sional formulas  in  terms  of  mass,  length,  and  time.  We  would  have 
obtained  the  following  formulation. 

Name  of  Quantity.  Symbol.  Dimensional  Formula. 

Charge,  e 

Field  strength,  E 

Energy  density,  u 


Again  we  are  to  seek  for  a  relation  between  the  energy  density 
and  the  field  strength.  Now  here  we  have  two  variables  and  three 
kinds  of  fundamental  units,  so  that  again  the  general  rule  would 
allow  no  dimensionless  products,  and  no  relation,  but  the  relation 
between  the  exponents  is  such  that  the  dimensionless  product  does 
exist,  and  in  fact  is  found  to  be  exactly  the  same  as  before.  The 
new  formulation  in  terms  of  different  fundamental  units  does  not 
change  the  result,  as  it  should  not. 

Many  persons  will  object  to  the  dimensional  formulas  given  for 
these  electrostatic  quantities  on  the  ground  that  we  arbitrarily  put 
the  dielectric  constant  of  empty  space  equal  to  unity,  whereas  we 
know  nothing  about  its  nature,  and  therefore  have  suppressed  cer- 
tain dimensions  which  are  essential  to  a  complete  statement  of  the 
problem.  This  point  of  view  will  of  course  not  be  disturbing  to  the 
reader  of  this  exposition,  who  has  come  to  see  that  there  is  nothing 


EXAMPLES  OF  DIMENSIONAL  ANALYSIS  79 

absolute  about  dimensions,  but  -that  they  may  be  anything  consistent 
with  a  set  of  definitions  which  agree  with  the  experimental  facts. 
However,  let  us  by  actual  example  carry  through  this  problem, 
including  the  dielectric  constant  of  empty  space  explicitly  as  a  new 
kind  of  fundamental  quantity  which  cannot  be  expressed  in  terms 
of  mass,  length,  and  time.  Call  the  dielectric  constant  of  empty 
space  k,  and  use  the  same  letter  to  stand  for  the  quantity  itself,  and 
its  dimension.  Then  the  unit  of  electrostatic  charge  is  now  defined 
by  the  relation,  force  =  e2/k  r2.  Field  strength  is  to  be  defined  as 
before  as  eE  =  Force.  If  we  formulate  the  problem  in  terms  of 
these  fundamentals,  the  electrostatic  field  equations  will  now  con- 
tain k  explicitly,  so  that  the  dimensional  constant  k  appears  in  the 
list  of  variables.  The  formulation  of  the  problem  is  now  as  follows : 

Name  of  Quantity.  Symbol.  Dimensional  Formula. 

Charge,  e 

Field  strength,  E 

Energy  density,  u 

Dielectric  constant  of  empty 

space,  k  k 

We  again  look  for  a  dimensionless  product  in  which  the  terms  are 
E,  u,  and  k,  and  find  the  result  to  be 

u  =  Const  k  E2. 

This  reduces  to  that  previously  found  on  putting  k  equal  to  unity, 
which  was  the  value  of  k  in  the  previous  formulation  of  the  problem. 
The  form  above  appears  somewhat  more  general  than  the  form 
previously  obtained  in  virtue  of  the  factor  k,  but  this  factor  does 
not  tell  us  anything  more  about  nature,  but  merely  shows  how  the 
formal  expression  of  the  result  will  change  when  we  change  the  for- 
mulation of  the  definitions  at  the  basis  of  our  system  of  equations. 
The  inclusion  of  the  factor  k  in  the  result  and  in  the  definitions  is 
therefore  of  no  advantage  to  us,  and  never  can  be  of  advantage,  if 
our  considerations  are  correct. 

There  has  been  much  written  on  the  ' '  true ' '  dimensions  of  k,  and 
much  speculation  about  the  various  physical  pictures  of  the  me- 
chanical structure  of  the  ether  which  follow  from  one  or  another 
assumption  as  to  the  ' '  true ' '  dimensions,  but  so  far  as  I  am  aware, 
no  result  has  been  ever  suggested  by  this  method  which  has  led  to 
the  discovery  of  new  facts,  although  it  cannot  be  denied  that  a  num- 


80  DIMENSIONAL  ANALYSIS 

ber  of  experiments  have  been  suggested  by  these  considerations,  as 
for  example  those  of  Lodge  on  the  mechanical  properties  of  the 
ether. 

KEFEBENCES 

(1)  Rayleigh,  Nat.  95,  66,  1915. 

(2)  Rayleigh,  Phil.  Mag.  41,  107  and  274,  1871. 

A  number  of  miscellaneous  examples  will  be  found  treated  in  a 
paper  by  C.  Runge,  Phys.  ZS.  17,  202,  1916. 

Speculations  as  to  the  " correct"  dimensions  to  be  given  to  elec- 
trical quantities,  and  deductions  as  to  the  properties  of  the  ether 
are  found  in  the  following  papers : 

A.  W.  Riicker,  Phil.  Mag.  27,  104,  1889. 
W.  Williams,  Phil.  Mag.  34,  234,  1892. 
O.  Lodge,  Phil.  Mag.  Nov.,  1882. 

Nat.  July  19,  1888. 

"Modern  Views  of  Electricity,"  Appendix. 
R.  A.  Fessenden,  Phys.  Rev.  10,  1  and  83,  1900. 
K.  R.  Johnson,  Phys.  ZS.  5,  635,  1904. 
A.  C.  Crehore,  Phys.  Rev.  14,  440,  1919. 

G.  F.  Fitzgerald,  Phil.  Mag.  27,  323,  1889'.  This  article  is  quoted 
entire. 

' '  Some  attention  has  lately  been  called  to  the  question  of  the 
dimensions  of  the  electromagnetic  units,  but  the  following  sug- 
gestion seems  to  have  escaped  notice. 

"The  electrostatic  system  of  units  may  be  defined  as  one  in 
which  the  electric  inductive  capacity  is  assumed  to  have  zero 
dimensions,  and  the  electromagnetic  system  is  one  in  which  the 
magnetic  inductive  capacity  is  assumed  to  have  zero  dimen- 
sions. Now  if  we  take  a  system  in  which  the  dimensions  of  both 
these  quantities  are  the  same,  and  of  the  dimensions  of  a  slow- 
ness, i.e.,  the  inverse  of  a  velocity  (T/L),  the  two  systems  be- 
come identical,  as  regards  dimensions,  and  differ  only  by  a 
numerical  coefficient,  just  as  centimeters  and  kilometers  do. 
There  seems  a  naturalness  in  this  result  which  justifies  the 
assumption  that  these  inductive  capacities  are  really  of  the 
nature  of  a  slowness.  It  seems  possible  that  they  are  related 
to  the  reciprocal  of  the  square  root  of  the  mean  energy  of 
turbulence  of  the  ether. ' ' 


CHAPTER  VII 

APPLICATIONS  OF  DIMENSIONAL  ANALYSIS  TO 
MODEL  EXPERIMENTS.  OTHER  ENGINEER- 
ING APPLICATIONS 

HITHERTO  we  have  applied  dimensional  analysis  to  problems  which 
could  be  solved  in  other  ways,  and  therefore  have  been  able  to  check 
our  results.  There  are,  however,  in  engineering  practise  a  large 
number  of  problems  so  complicated  that  the  exact  solution  is  not 
obtainable.  Under  these  conditions  dimensional  analysis  enables  us 
to  obtain  certain  information  about  the  form  of  the  result  which 
could  be  obtained  in  practise  only  by  experiments  with  an  impos- 
sibly wide  variation  of  the  arguments  of  the  unknown  function.  In 
order  to  apply  dimensional  analysis  we  merely  have  to  know  what 
kind  of  a  physical  system  it  is  that  we  are  dealing  with,  and  what 
the  variables  are  which  enter  the  equation ;  we  do  not  even  have  to 
write  the  equations  down  explicitly,  much  less  solve  them.  In  many 
cases  of  this  sort,  the  partial  information  given  by  dimensional 
analysis  may  be  combined  with  measurements  on  only  a  part  of  the 
totality  of  physical  systems  covered  by  the  analysis,  so  that  together 
all  the  information  needed  is  obtained  with  much  less  trouble  and 
expense  than  would  otherwise  be  possible.  This  method  is  coming 
to  be  of  more  and  more  importance  in  technical  studies,  and  has 
recently  received  a  considerable  impetus  from  the  necessities  of 
airplane  design.  The  method  has  received  wide  use  at  the  National 
Physical  Laboratory  in  England,  and  at  the  Bureau  of  Standards 
in  this  country,  and  has  been  described  in  numerous  papers.  At  the 
Bureau  of  Standards  Dr.  Edgar  Buckingham  has  been  largely  in- 
strumental in  putting  the  results  of  dimensional  analysis  into  such 
a  form  that  they  may  be  easily  applied,  and  in  making  a  number 
of  important  applications. 

The  nature  of  the  results  obtainable  by  this  method  may  be  illus- 
trated by  a  very  simple  example.  Suppose  that  it  is  desired  to  con- 
struct a  very  large  and  expensive  pendulum  of  accurately  prede- 
termined time  of  swing.  Dimensional  analysis  shows  that  the  time 


82  DIMENSIONAL  ANALYSIS 

of  oscillation  of  the  entire  system  of  all  pendulums  is  given  by  the 
formula  t  =  Const  VVg-  Hence  to  determine  the  time  of  any 
pendulum  whatever,  it  is  sufficient  to  determine  by  experiment  only 
the  value  of  the  constant  in  the  equation.  The  constant  may  evi- 
dently be  found  by  a  single  experiment  on  a  pendulum  of  any 
length  whatever.  The  experimental  pendulum  may  be  made  as 
simple  as  we  please,  and  by  measuring  the  time  of  swing  of  it,  the 
time  of  swing  of  the  projected  large  pendulum  may  be  obtained. 

The  case  of  the  pendulum  is  especially  simple  in  that  no  arbitrary 
function  appeared  in  the  result.  Now  let  us  consider  the  more  gen- 
eral case  which  may  be  complicated  by  the  appearance  of  an  arbi- 
trary function.  Suppose  that  the  variables  of  the  problem  are 
denoted  by  Q1?  Q2,  etc.,  and  that  the  dimensionless  products  are 
found,  and  that  the  result  is  thrown  into  the  form 

Q,  =  Q:-Q?> *(Q?Qf-     -,Q?Qf-      -) 

where  the  arguments  of  the  function  and  the  factor  outside  embrace 
all  the  dimensionless  products,  so  that  the  result  as  shown  is  general. 
Now  in  passing  from  one  physical  system  to  another  the  arbitrary 
function  will  in  general  change  in  an  unknown  way,  so  that  little 
if  any  useful  information  could  be  obtained  by  indiscriminate 
model  experiments.  But  if  the  models  are  chosen  in  such  a  restricted 
way  that  all  the  arguments  of  the  unknown  function  have  the  same 
value  for  the  model  as  for  the  full  scale  example,  then  the  only 
variable  in  passing  from  model  to  full  scale  is  in  the  factors  outside 
the  functional  sign,  and  the  manner  of  variation  of  these  factors 
is  known  from  the  dimensional  analysis. 

Two  systems  which  are  so  related  to  each  other  that  the  argu- 
ments inside  the  unknown  functional  sign  are  equal  numerically 
are  said  to  be  physically  similar  systems. 

It  is  evident. that  a  model  experiment  can  give  valuable  informa- 
tion if  the  model  is  constructed  in  such  a  way  that  it  is  physically 
similar  to  the  full  scale  example.  The  condition  of  physical  simi- 
larity involves  in  general  not  only  conditions  on  the  dimensions  of 
the  model  but  on  all  the  other  physical  variables  as  well. 
'As  an  example  let  us  consider  the  resistance  experienced  by  a 
body  of  some  definable  shape  in  moving  through  an  infinite  mass  of 
fluid.  Special  cases  of  this  problem  are  the  resistance  encountered 
by  a  projectile,  by  an  airplane,  by  a  submarine  in  deep  water,  or  by 
a  falling  raindrop.  The  problem  is  evidently  one  of  mechanics,  and 


APPLICATIONS  TO  MODEL  EXPERIMENTS  83 

involves  the  equations  of  hydrodynamics.  The  conditions  are  ex- 
ceedingly complicated,  and  would  be  difficult  to  formulate  in  pre- 
cise mathematical  terms,  but  we  perhaps  may  imagine  it  done  by 
some  sort  of  a  super-being.  The  important  thing  to  notice  is  that  no 
dimensional  constants  appear  in  the  equation  of  hydrodynamics  if 
the  ordinary  mechanical  units  in  terms  of  mass,  length,  and  time 
are  used,  so  that  the  result  will  involve  only  the  measurable  physical 
variables.  The  variables  are  the  resistance  to  the  motion,  the  velocity 
of  motion,  the  shape  of  the  body,  which  we  may  suppose  specified 
by  some  absolute  dimension  and  the  ratio  to  it  of  certain  other 
lengths  (as,  for  instance,  the  shape  of  an  ellipsoid  may  be  specified 
by  the  length  of  the  longest  axis  and  the  ratio  to  this  axis  of  the 
other  axes)  and  the  constants  of  the  fluid,  which  are  its  density, 
viscosity,  and  compressibility,  the  latter  of  which  we  may  specify 
by  giving  the  velocity  of  sound  in  the  fluid.  We  suppose  that  gravity 
does  not  enter  the  results,  that  is,  the  body  is  in  uniform  motion  at 
a  constant  level,  so  that  no  work  is  done  by  the  gravitational  forces. 
The  formulation  of  the  problem  is  now  as  follows. 

Name  of  Quantity.                 Symbol.  Dimensional  Formula. 

Resistance,  R  MLT~2 

Velocity,                                           v  LT^1 

Absolute  dimension,                        1  L 

Density  of  fluid,                               d  ML~3 

Viscosity  of  fluid,                            p  ML^T-1 

Velocity  of  sound  in  the  fluid,  v'  LT"1 
Shape  factors,  fixing  shape  of 

body,  r±,  r2,  etc.         .      0 

We  have  here  six  variables,  not  counting  the  shape  factors,  which 
may  have  any  number  depending  on  the  geometrical  complexity  of 
the  body,  so  that  there  are  three  dimensionless  products  exclusive 
of  the  shape  factors,  which  are  already  dimensionless.  One  of  these 
three  dimensionless  products  is  obvious  on  inspection,  and  is  v'/v. 
We  have  to  find  the  other  dimensionless  products  in  the  way  best 
adapted  to  this  particular  problem.  Since  we  are  interested  in  the 
resistance  to  the  motion,  we  choose  this  as  the  term  with  unit 
exponent  in  one  of  the  products,  so  that  we  may  write  the  result 
with  R  standing  alone  on  the  left-hand  side  of  the  equation.  We  find 
by  the  methods  that  we  have  used  so  many  times  that  there  are  two 


84  DIMENSIONAL  ANALYSIS 

dimensionless  products  of  the  forms  R  v~2  1~2  d"1  and  /u,  v"1  1"1  d"1, 
so  that  the  final  solution  takes  the  form 

R  =  v2  12  d  f  (^/v  1  d,  v'/v,  r±,  r2,  -       -)  . 

This  formula  is  so  broad  as  to  cover  a  wide  range  of  experimental 
conditions.  If  the  velocity  is  low,  the  problem  reduces  to  one  of 
equilibrium  in  which  the  forces  on  the  solid  body  immersed  in  the 
fluid  are  held  in  equilibrium  by  the  forces  due  to  the  viscosity  in  the 
fluid.  The  resistance  does  not  depend  on  the  density  of  the  fluid,  nor 
on  the  velocity  of  sound  in  it.  Evidently  if  the  density  is  to  disap- 
pear from  the  above  result,  the  argument  /u/vld  must  come  outside 
the  functional  sign  as  a  factor,  and  for  slow  motion  the  law  of 
resistance  takes  the  form 

Rr=vl/xf  (r±,r2,  ----  ). 

The  resistance  at  low  velocities  is  therefore  proportional  to  the  vis- 
cosity, to  the  velocity,  and  to  the  linear  dimensions,  and  besides  this 
depends  only  on  the  geometrical  shape  of  the  body.  We  have  already 
met  a  special  case  in  the  Stoke  's  problem  of  the  sphere,  in  the  intro- 
ductory chapter. 

For  a  domain  of  still  higher  velocities  the  density  of  the  fluid 
plays  an  important  part,  since  some  of  the  force  acting  on  the  body 
is  due  to  the  momentum  carried  away  from  the  surface  of  the  body 
by  the  fluid  in  the  form  of  eddies  (and  the  momentum  carried  away 
obviously  depends  on  the  density  of  the  fluid),  but  the  velocity  of 
sound  has  not  yet  begun  to  affect  the  result,  which  means  that  the 
fluid  acts  sensibly  like  an  incompressible  liquid.  This  is  the  realm 
of  velocities  of  interest  in  airplane  work.  Under  these  conditions  the 
argument  v'/v  drops  out  of  the  function,  therefore,  and  the  result 
becomes 


Let  us  stop  to  inquire  how  the  information  given  by  this  equation 
can  be  used  in  devising  model  experiments.  What  we  desire  to  do  is 
to  make  a  measurement  of  the  resistance  encountered  by  the  model 
under  certain  conditions,  and  to  infer  from  this  what  would  be  the 
resistance  encountered  by  the  full  size  example.  It  is  in  the  first 
place  obvious  that  the  unknown  function  must  have  the  same  value 
for  the  model  and  the  original.  This  means,  since  the  function  is 
entirely  unknown,  that  all  the  arguments  must  have  the  same  value 


APPLICATIONS  TO  MODEL  EXPERIMENTS  85 

for  the  model  and  the  original.  rx,  r2,  etc.,  must  therefore  be  the 
same  for  both,  or  in  other  words,  the  model  and  the  original  must  be 
geometrically  of  the  same  shape.  Furthermore,  ju/vld  must  have  the 
same  numerical  value  for  both.  If  the  model  experiment  is  to  be  per- 
formed in  air,  as  it  usually  is,  /A  and  d  are  the  same  for  the  model 
and  original,  so  that  vl  must  be  the  same  for  model  and  original. 
That  is,  if  the  model  is  one-tenth  the  linear  dimensions  of  the  orig- 
inal, then  its  velocity  must  be  ten  times  as  great  as  that  of  the 
original.  Under  these  conditions  the  formula  shows  that  the  resist- 
ance encountered  by  the  model  is  exactly  the  same  as  that  encoun- 
tered by  the  original.  Now  this  requirement  imposes  such  difficult 
conditions  to  meet  in  practise,  demanding  velocities  in  the  model 
of  the  order  of  thousands  of  miles  per  hour,  that  it  would  seem  at 
first  sight  that  we  had  proved  the  impossibility  of  model  experi- 
ments of  this  sort.  But  in  practise  the  function  of  //./v  1  d  turns 
out  to  have  such  special  properties  that  much  important  informa- 
tion can  nevertheless  be  obtained  from  the  model  experiment.  If 
measurements  are  made  on  the  resistance  of  the  model  at  various 
speeds,  and  the  corresponding  values  of  the  function  calculated 
(that  is,  if  the  measured  resistances  are  divided  by  v2 12  d),  it  will  be 
found  that  at  high  values  of  the  velocity  the  function  f  approaches 
asymptotically  a  constant  value.  This  means  that  at  high  velocities 
the  resistance  approaches  proportionality  to  the  square  of  the  veloc- 
ity. It  is  sufficient  to  carry  the  experiment  on  .the  model  only  to 
such  velocities  that  the  asymptotical  value  of  the  function  may  be 
found,  in  order  to  obtain  all  the  information  necessary  about  the 
behavior  of  the  full  scale  example,  for  obviously  we  now  know  that 
the  resistance  is  proportional  to  the  square  of  the  velocity,  and  the 
model  experiment  has  given  the  factor  of  proportionality.  The  only 
doubtful  point  in  this  proposed  procedure  is  the  question  as  to 
whether  it  is  possible  to  reach  with  the  model  speeds  high  enough 
to  give  the  asymptotic  value,  and  this  question  is  answered  by  the 
actual  experiment  in  the  affirmative. 

The  fact  that  at  the  velocities  of  actual  airplane  work  the-  resist- 
ance has  become  proportional  to  the  square  of  the  velocity  means, 
according  to  the  analysis,  that  the  viscosity  no  longer  plays  a 
dominant  part.  This  means  that  skin  friction  has  dropped  out  as  an 
important  part  of  the  retarding  force,  and  that  all  the  work  of 
driving  the  airplane  is  used  in  creating  eddies  in  the  air.  Freedom 
from  viscosity  and  complete  turbulence  of  motion  are  seen  by  the 


86  DIMENSIONAL  ANALYSIS 

analysis  to  be  the  same  thing.  This  view  of  the  phenomena  is  abun- 
dantly verified  by  experiment. 

Let  us  consider  the  possibility  of  making  model  experiments  in 
some  other  medium  than  air.  If  we  choose  water  as  the  medium  for 
the  model  we  must  so  choose  the  dimensions  and  the  velocity  of  the 
model  that  /x/v  d  1  for  the  model  is  equal  to  p/v  d  1  for  the  origi- 
nal. Now  for  water  /x  is  10~2  and  d  is  1,  whereas  for  air  //,  is  170  X 
10~6  and  d  is  1.29  X  10~3.  Substituting  these  values  shows  that  vl 
for  the  model  must  be  about  one-thirteenth  of  the  value  for  the 
original.  As  a  factor  one-thirteenth  is  itself  about  the  reduction  in 
size  that  would  be  convenient  for  the  model ;  this  would  mean  that 
the  model  in  water  must  travel  at  about  the  same  rate  as  the  original 
in  air.  Such  high  velocities  in  water  are  difficult,  and  there  seems 
no  advantage  in  using  water  over  the  actual  procedure  that  is  pos- 
sible in  air. 

Consider  now  still  higher  velocities,  such  as  those  of  a  projectile, 
which  may  be  higher  than  the  velocity  of  sound  in  the  medium,  so 
that  the  medium  has  difficulty  in  getting  out  of  the  way  of  the  body, 
and  we  have  a  still  different  order  of  effects.  At  these  velocities  the 
viscosity  has  entirely  disappeared  from  the  result,  which  now  takes 
the  form 

R  =  v2l2df  (v'/v,ri,r2, ). 

If  we  are  now  to  make  model  experiments,  it  is  evident  that  the 
model  projectile  must  be  of  the  same  shape  as  the  original,  and 
furthermore  that  v'/v  must  have  the  same  value  for  the  model  and 
the  original.  If  the  model  experiment  is  ,made  in  air,  v'  for  the 
model  will  be  the  same  as  for  the  original,  so  that  v  must  be  the  same 
also.  That  is,  the  original  and  the  model  must  travel  at  the  same 
speed.  Under  these  conditions  the  formula  shows  that  the  resistance 
varies  as  the  area  of  cross  section  of  the  projectile.  The  requirement 
that  the  model  must  travel  at  the  same  speed  as  the  original  imposes 
such  severe  restrictions  in  practise  that  model  experiments  on  pro- 
jectiles are  most  difficult,  and  apparently  have  not  yet  been  success- 
fully made,  but  all  the  information  has  been  derived  from  experi- 
ence with  actual  projectiles. 

We  may  try  to  avoid  the  difficulty  by  making  the  experiment  in 
another  medium,  such  as  water.  But  the  velocity  of  sound  in  water 
is  of  the  order  of  five  times  that  in  air,  so  that  the  conditions  would 
require  that  the  velocity  of  the  model  projectile  in  water  should  be 


APPLICATIONS  TO  MODEL  EXPERIMENTS  87 

five  times  that  of  the  actual  projectile  in  air,  an  impossible  require- 
ment. 

Besides  these  applications  to  model  experiments,  the  results  of 
dimensional  analysis  may  be  applied  in  other  branches  of  engineer- 
ing. At  the  Bureau  of  Standards  extensive  applications  have  been 
made  in  discussing  the  performance  of  various  kinds  of  technical 
instruments.  A  class  of  instruments  for  the  same  purpose  have 
certain  characteristics  in  common  so  that  it  is  often  possible  to  write 
down  a  detailed  analysis  applicable  to  all  instruments  of  the  par- 
ticular type.  Dimensional  analysis  gives  certain  information  about 
what  the  result  of  such  an  analysis  must  be,  so  that  it  is  possible  to 
make  inferences  from  the  behavior  of  one  instrument  concerning  the 
behavior  of  other  instruments  of  somewhat  different  construction. 
This  subject  is  treated  at  considerable  length  and  a  number  of 
examples  are  given  in  Aeronautic  Instruments  Circular  No.  30  of 
the  Bureau  of  Standards,  written  by  Mr.  M.  D.  Hersey. 

BEFEKENCES 

Applications  of  dimensional  analysis  to  subjects  of  a  technical 
character  will  be  found  in  the  following  papers : 
0.  Reynolds,  Phil.  Trans.  Roy.  Soc.  174,  935,  1883. 
Rayleigh,  Phil.  Mag.  34,  59,  1892,  and  8,  66,  1904. 
E.  Buckingham,  Phys.  Rev.  4,  345,  1914. 

Trans.  Amer.  Soc.  Mech.  Eng.  37,  263,  1915. 

Engineering,  March  13,  1914. 

E.  Buckingham  and  J.  D.  Edwards,  Sci.  Pap.  Bur.  Stds.  No.  359, 
1920. 

M.  D.  Hersey,  Jour.  Wash.  Acad.  6,  569,  1916. 
Sci.  Pap.  Bur.  Stds.  No.  331,  1919. 

Aeronautical  Instruments  Circulars  of  the  Bureau  of  Stand- 
ards, No.  30,  1919,  and  No.  32,  1918. 

W.  L.  Cowley  and  H.  Levy,  Aeronautics  in  Theory  and  Experi- 
ment, Longmans,  Green  and  Co.,  1918,  Chap.  IV. 

E.  B.  Wilson,  Aeronautics,  John  Wiley  and  Sons,  1920,  Chap.  XI. 


CHAPTER  VIII 
APPLICATIONS  TO  THEORETICAL  PHYSICS 

THE  methods  of  dimensional  analysis  are  worthy  of  playing  a  much 
more  important  part  as  a  tool  in  theoretical  investigation  than  has 
hitherto  been  realized.  No  investigator  should  allow  himself  to  pro- 
ceed to  the  detailed  solution  of  a  problem  until  he  has  made  a 
dimensional  analysis  of  the  nature  of  the  solution  which  will  be 
obtained,  and  convinced  himself  by  appeal  to  experiment  that  the 
points  of  view  embodied  in  the  underlying  equations  are  sound. 

Probably  one  difficulty  that  has  been  particularly  troublesome  in 
theoretical  applications  has  been  the  matter  of  dimensional  con- 
stants ;  it  is  in  just  such  theoretical  investigations  that  dimensional 
constants  are  most  likely  to  appear,  and  with  no  clear  conception 
of  the  nature  of  a  dimensional  constant  or  when  to  expect  its  ap- 
pearance, hesitancy  is  natural  in  applying  the  method.  But  after 
the  discussion  of  the  preceding  pages,  the  matter  of  dimensional 
constants  should  now  be  readily  handled  in  any  special  problem. 

The  indeterminateness  of  the  numerical  factors  of  proportionality 
is  often  also  felt  to  be  a  disadvantage  of  the  dimensional  method, 
but  in  many  theoretical  investigations  it  is  often  possible  to  obtain 
approximate  information  about  the  numerical  order  of  magnitude 
of  the  results.  Our  considerations  with  regard  to  dimensional 
analysis  show  that  any  numerical  coefficients  in  the  final  result  are 
the  result  of  mathematical  operations  performed  on  the  original 
equations  of  motion  (in  the  general  sense)  of  the  system.  Now  it  is 
a  result  of  general  observation  that  such  mathematical  operations 
usually  do  not  introduce  any  very  large  numerical  factors,  or  any 
very  small  ones.  Any  very  large  or  small  numbers  in  our  equations 
almost  always  are  the  result  of  the  substitution  of  the  numerical 
value  of  some  physical  quantity,  such  as  the  number  of  atoms  in  a 
cubic  centimeter,  or  the  electrostatic  charge  on  the  electron,  or  the 
velocity  of  light.  Accordingly,  if  the  analysis  is  carried  through 
with  all  the  physical  quantities  kept  in  literal  form,  we  may  expect 
that  the  numerical  coefficients  will  not  be  large  or  small. 

This  observation  may  be  used  conversely.  Suppose  that  we  suspect 


APPLICATIONS  TO  THEORETICAL  PHYSICS         89 

a  connection  between  certain  quantities,  but  as  yet  do  not  know 
enough  of  the  nature  of  the  physical  system  to  be  able  to  write  down 
the  equations  of  connection,  or  even  to  be  sure  what  would  be  the 
elements  which  would  enter  an  equation  of  connection.  We  assume 
that  there  is  a  relation  between  certain  quantities,  and  then  by  a 
dimensional  analysis  find  what  the  form  of  the  relation  must  be.  We 
then  substitute  into  the  relation  the  numerical  values  of  the  physi- 
cal quantities,  and  thus  get  the  numerical  value  of  the  unknown 
coefficient.  If  this  coefficient  is  of  the  order  of  unity  (which  may 
mean  not  of  the  order  of  1010  according  to  our  sanguinity)  the  sus- 
pected relation  appears  as  not  intrinsically  improbable,  and  we 
continue  to  think  about  the  matter  to  discover  what  the  precise 
relation  between  the  elements  may  be.  If,  on  the  other  hand,  the 
coefficient  turns  out  to  be  large  or  small,  we  discard  the  idea  as 
improbable. 

An  exposition  of  this  method,  and  an  interesting  example  were 
given  by  Einstein1  in  the  early  days  of  the  study  of  the  specific 
heats  of  solids  and  their  connection  with  quantum  phenomena.  The 
question  was  whether  the  same  forces  between  the  atoms  which 
determine  the  ordinary  elastic  behavior  of  a  solid  might  not  also  be 
the  forces  concerned  in  the  infra  red  characteristic  optical  fre- 
quencies. This  view  evidently  had  important  bearings  on  our  whole 
conception  of  the  nature  of  the  forces  in  a  solid,  and  the  nature  of 
optical  and  thermal  oscillations. 

For  the  rough  analysis  of  the  problem  in  these  terms  we  may 
regard  the  solid  as  an  array  of  atoms  regularly  spaced  at  the  corners 
of  cubes.  In  our  analysis  we  shall  evidently  want  to  know  the  mass 
of  the  atoms,  and  their  distance  apart  (or  the  nunjber  per  cm3). 
Furthermore,  if  our  view  of  the  nature  of  the  forces  is  correct  the 
nature  of  the  forces  between  the  atoms  is  sufficiently  characterized 
by  an  elastic  constant,  which  we  will  take  as  the  compressibility. 
These  elements  should  now  be  sufficient  to  determine  the  infra  red 
characteristic  frequency.  We  make  our  usual  analysis  of  the 
problem. 

Name  of  Quantity.  Symbol.  Dimensional  Formula. 
Characteristic  frequency,  v  T"1 

Compressibility,  k  M-1LT2 

Number  of  atoms  per  cm8,  N  L~3 

Mass  of  the  atom,  m  M 


90  DIMENSIONAL  ANALYSIS 

There  should  be  one  dimensionless  product  in  these  quantities, 
and  it  is  at  once  found  to  be  k  v2  N*  m.  The  final  result  is  therefore 

k  =  Const  v~2  N-S  m-1. 

"We  now  take  the  numerical  values  pertaining  to  some  actual  sub- 
stance and  substitute  in  the  equation  to  find  the  numerical  value  of 
the  coefficient.  For  copper,  k  =  7  X  10~13,  v  —  7.5  X  1012,  m  =  1.06 
X  10~22,  and  N  =  7.5  X  1022.  Substituting  these  values  gives  for 
the  constant  0.18.  This  is  of  the  order  of  unity,  and  the  point  o'f 
view  is  thus  far  justified.  It  is  of  course  now  a  matter  of  history 
that  this  point  of  view  is  the  basis  of  Debye  's  analysis  of  the  specific 
heat  phenomena  in  a  solid,  and  that  it  is  brilliantly  justified  by 
experiment. 

Another  example  of  this  sort  of  argument  concerning  the  magni- 
tude of  the  constants  is  given  by  Jeans.2  The  question  was  whether 
the  earth  has  at  any  time  in  its  past  history  passed  through  a  stage 
of  gravitational  instability,  and  whether  this  instability  has  had  any 
actual  relation  to  the  course  of  evolution.  A  preliminary  examina- 
tion by  the  method  of  dimensions  showed  Jeans  what  must  be  the 
form  of  the  relation  between  the  variables  such  as  mean  density, 
radius,  elastic  constants,  etc.,  at  the  moment  of  gravitational  insta- 
bility, and  then  a  substitution  of  the  numerical  values  for  the  earth 
gave  a  coefficient  of  the  order  of  unity.  This  preliminary  examina- 
tion showed,  therefore,  that  it  was  quite  conceivable  that  gravita- 
tional instability  might  be  a  factor  at  some  time  past  or  future  in 
the  earth 's  history,  and  a  more  detailed  examination  of  the  problem 
was  accordingly  undertaken. 

Consider  another  application  of  the  same  argument,  this  time 
with  a  negative  result.  Let  us  suppose  that  we  are  trying  to  con- 
struct an  electrodynamic  theory  of  gravitation,  and  that  we  regard 
the  gravitational  field  as  in  some  way,  as  yet  undiscovered,  con- 
nected with  the  properties  of  the  electron,  to  be  deduced  by  an 
application  of  the  field  equations  of  electrodynamics.  Now  in  the 
field  equations  there  occurs  a  dimensional  constant  c,  the  ratio  of 
the  electromagnetic  and  electrostatic  units,  which  is  known  to  be 
of  the  dimensions  of  a  velocity,  and  numerically  to  be  the  same  as 
the  velocity  of  light. 

In  searching  for  a  relation  of  the  sort  suspected,  we  therefore 
consider  as  the  variables  the  charge  on  the  electron,  the  mass  of  the 


APPLICATIONS  TO  THEORETICAL  PHYSICS         91 

electron  (for  charge  and  mass  together  characterize  the  electron), 
the  velocity  of  light,  and  the  gravitational  constant.  We  make  the 
following  analysis : 

Name  of  Quantity.  Symbol.  Dimensional  Formula. 

Gravitational  constant,  G 

Charge  on  electron,  e 

Mass  of  electron,  m  M 

Velocity  of  light,  c  LT~* 

We  have  four  variables,  and  three  fundamental  units,  so  that  we 
expect  one  dimensionless  product.  This  is  at  once  found  to  be 
Gm2e~2,  the  velocity  of  light  not  entering  the  hypothetical  relation, 
and  the  final  result  taking  the  significantly  simple  form 

G=r  Const  (e/m)2. 

We  now  substitute  numerical  values  to  find  the  magnitude  of  the 
constant.  G  =  6.658  X  10~8,  and  e/m  =  5.3  X  1017,  so  that  Const 
=  2.35  X  10~43. 

The  constant  is  seen  therefore  to  be  impossibly  small,  and  we  give 
up  the  attempt  to  think  how  there  might  be  a  relation  between  these 
quantities,  although  the  simplicity  of  the  dimensional  relation  be- 
tween G  and  e/m  is  arresting. 

Identity  of  dimensional  formulas  must  not  be  thought,  therefore, 
to  indicate  an  a  priori  probability  of  any  sort  of  physical  relation. 
When  there  are  so  many  kinds  of  different  physical  quantities  ex- 
pressed in  terms  of  a  few  fundamental  units,  there  cannot  help 
being  all  sorts  of  accidental  relations  between  them,  and  without 
further  examination  we  cannot  say  whether  a  dimensional  relation 
is  real  or  accidental.  Thus  the  mere  fact  that  the  dimensions  of  the 
quantum  are  those  of  angular  momentum  does  not  justify  us  in 
expecting  that  there  is  a  mechanism  to  account  for  the  quantum 
consisting  of  something  or  other  in  rotational  motion. 

The  converse  of  the  theorem  attempted  above  does  hold,  however. 
If  there  is  a  true  physical  connection  between  certain  quantities, 
then  there  is  also  a  dimensional  relation.  This  result  may  be  used 
to  advantage  as  a  tool  of  exploration. 

Consider  now  a  problem  showing  that  any  true  physical  relation 
must  also  involve  a  dimensional  relation.  Suppose  that  we  are  try- 


92  DIMENSIONAL  ANALYSIS 

ing  to  build  up  a  theory  of  thermal  conduction,  and  are  searching 
for  a  connection  between  the  mechanism  of  thermal  conduction  and 
the  mechanism  responsible  for  the  ordinary  thermodynamic  be- 
havior of  substances.  The  thermodynamic  behavior  may  be  consid- 
ered as  specified  by  the  compressibility,  thermal  expansion,  specific 
heat  (all  taken  per  unit  volume),  and  the  absolute  temperature.  If 
only  those  aspects  of  the  mechanism  which  are  responsible  for  the 
thermodynamic  behavior  are  also  effective  in  determining  the 
thermal  conductivity,  then  it  must  be  possible  to  find  a  dimensional 
relation  between  the  thermodynamic  elements  and  the  thermal  con- 
ductivity. We  have  the  following  formulation  of  the  problem. 

Name  of  Quantity.  Symbol.  Dimensional  Formula. 

Thermal  conductivity,  /x 
Compressibility  per  unit  vol- 

ume, k  M~2L4T2 

Thermal   expansion   per   unit 

volume,  A  Mr^L3^1 

Specific  heat  per  unit  volume,  C  L2^2^1 

Absolute  temperature,  0  0 

We  are  to  seek  for  a  dimensionless  product  in  these  variables. 
There  are  five  variables,  and  four  fundamental  kinds  of  quantity, 
so  that  we  would  expect  one  dimensionless  product.  Since  /*  is  the 
quantity  in  which  we  are  particularly  interested,  we  choose  it  as  the 
member  of  the  product  with  unity  for  the  exponent,  write  the 
product  in  the  form 


and  attempt  to  solve  for  the  exponents  in  the  usual  way.  We  soon 
encounter  difficulties,  however,  for  it  appears  that  the  equations  are 
inconsistent  with  each  other.  This  we  verify  by  writing  down  the 
determinant  of  the  exponents  in  the  dimensional  formulas  for  k,  A, 
C,  and  0.  The  determinant  is  found  to  vanish,  which  means  that  the 
dimensionless  product  does  not  exist.  Hence  the  hypothetical  rela- 
tion between  thermal  conductivity  and  thermodynamic  data  does 
not  exist,  and  the  mechanism  of  the  solid  must  have  other  properties 
than  those  sufficient  to  account  for  the  thermodynamic  data  alone. 
We  now  give  a  simple  discussion  of  the  problem  of  radiation  from 
a  black  body.  A  much  more  elaborate  discussion  has  been  given  by 


APPLICATIONS  TO  THEORETICAL  PHYSICS         93 

Jeans.3  The  paper  of  Jeans  is  also  interesting  because  he  uses  a 
system  of  electrical  units  in  which  the  dielectric  constant  of  empty 
space  is  introduced  explicitly.  It  is  easy  to  see  on  a  little  examina- 
tion that  he  would  have  obtained  the  same  result  with  a  system  of 
units  in  which  the  dielectric  constant  of  empty  space  is  defined  as 
unity. 

Let  us  now  consider  a  cavity  with  walls  which  have  absolutely 
no  specific  properties  of  their  own,  but  are  perfect  reflectors  of  any 
incident  radiation.  Inside  the  cavity  is  a  rarefied  gas  composed  of 
electrons.  If  the  gas  is  rarefied  enough  we  know  from  such  con- 
siderations as  those  given  by  Richardson  in  considering  thermionic 
emission  that  the  electrons  function  like  a  perfect  gas,  the  effect  of 
the  space  distribution  of  electrostatic  charge  being  negligible  in 
comparison  with  the  forces  due  to  collisions  as  gas  particles.  The 
electron  gas  in  the  cavity  is  to  be  maintained  at  a  temperature  0. 
The  electrons  are  acted  on  by  two  sets  of  forces;  the  collisional 
forces  with  the  other  electrons,  which  are  of  the  nature  of  the  forces 
between  atoms  in  ordinary  kinetic  theory,  and  the  radiational  field 
in  the  ether.  Since  the  electrons  are  continually  being  accelerated, 
they  are  continually  radiating,  and  they  are  also  continually  absorb- 
ing energy  from  the  radiational  field  of  the  ether.  The  system  must 
eventually  come  to  equilibrium  with  a  certain  energy  density  in  the 
ether,  the  electrons  possessing  at  the  same  time  the  kinetic  energy 
appropriate  to  gas  atoms  at  the  temperature  of  the  enclosure.  The 
detailed  solution  of  the  problem  obviously  involves  a  most  com- 
plicated piece  of  statistical  analysis,  but  a  dimensional  analysis 
gives  much  information  about  the  form  of  the  result. 

In  solving  this  problem  we  shall  have  to  use  the  field  equations 
of  electrodynamics,  so  that  the  velocity  of  light  will  be  a  dimensional 
constant  in  the  result.  The  charge  and  the  mass  of  the  electron  must 
be  considered,  the  absolute  temperature,  and  the  gas  constant,  be- 
s  cause  this  determines  the  kinetic  energy  of  motion  of  the  electrons 
as  a  function  of  temperature.  The  number  of  electrons  per  cm3  does 
not  enter,  because  we  know  from  kinetic  theory  that  the  mean 
velocity  of  the  electrons  is  independent  of  their  number.  The  second 
law  of  thermodynamics  also  shows  that  the  energy  density  in  the 
enclosure  is  a  function  of  the  temperature,  and  not  of  the  density 
of  the  electron  gas. 

Our  formulation  of  the  problem  is  now  as  follows : 


94  DIMENSIONAL  ANALYSIS 

Name  of  Quantity.  Symbol.  Dimensional  Formula. 

Energy  density,  u  ML"1 11"2 

Velocity  of  light,  c 

Mass  of  electron,  m  M 

Charge  of  electron,  e 

Absolute  temperature,  6  6 

Gas  constant,  k 

The  ordinary  electrostatic  system  of  units  is  used.  There  are  here 
six  variables  and  four  fundamental  kinds  of  unit,  hence  two  dimen- 
sionless  products,  unless  there  should  be  some  special  relation  be- 
tween the  exponents.  Since  we  are  especially  interested  in  u  we 
choose  this  as  the  member  of  one  of  the  products  with  unit  exponent. 
We  find  in  the  usual  way  that  two  products  are 

u  e6  k~4  0~4  and  k  0  m-1  c~2. 
The  result  therefore  takes  the  form 

u  =  k4  e-6  6*  f  (k  e  m-1  c~2). 

We  as  yet  know  nothing  of  the  nature  of  the  arbitrary  function. 
The  argument  of  the  function,  however,  is  seen  to  have  a  definite 
physical  significance,  kflrn"1  is  half  the  square  of  the  velocity  of  the 
electron  (k0  being  its  kinetic  energy),  so  that  the  argument  is  one- 
half  the  square  of  the  ratio  of  the  velocity  of  the  electron  to  the 
velocity  of  light.  Now  this  quantity  remains  exceedingly  small  in 
the  practical  range  of  temperature,  so  that  whatever  the  form  of  the 
function,  we  know  that  we  have  a  function  of  a  quantity  which  is 
always  small.  By  an  extension  of  the  reasoning  which  we  employed 
for  the  numerical  value  of  any  coefficients  to  be  met  with  in  dimen- 
sional analysis,  we  may  say  that  the  probability  is  that  the  numeri- 
cal value  of  such  a  function  is  sensibly  the  same  as  its  value  for  the 
value  zero  of  the  argument,  that  is,  the  function  may  be  replaced 
by  a  constant  for  the  range  of  values  of  the  variable  met  with  in 
practise.  Hence  with  much  plausibility  we  may  expect  the  result 
to  be  of  the  form 

u  =  Const  k4  e-6  0*. 

0  is  the  only  physical  variable  on  the  right-hand  side  of  this  equa- 
tion, so  that  as  far  as  physical  variables  go  the  result  may  be  written 
in  the  form 

u  =  a  P. 


APPLICATIONS  TO  THEORETICAL  PHYSICS         95 

This,  of  course,  is  the  well-known  Stefan 's  law,  which  checks  with 
experiment.  The  result  therefore  justifies  to  a  certain  extent  the 
views  which  led  us  to  the  result. 

Our  argument  about  the  size  of  numerical  coefficients  would  lead 
us  to  expect  that  the  constant  in  the  first  form  of  the  result  could 
not  be  too  large  or  too  small.  That  is,  if  we  put  a  =  Const  k*  e"6,  the 
result  should  have  a  certain  simplicity  of  form,  such  as  might  seem 
to  be  a  plausible  result  of  a  mathematical  operation.  Now  Lewis  and 
Adams4  have  called  attention  to  the  fact  that  within  the  limits  of 
experimental  error  the  constant  of  Stefan's  law  may  be  written  in 
the  form 

a  — k4/(47re)6. 

Although  (4  7r)6  is  not  an  especially  small  number  in  the  sense  of 
the  original  formulation  by  Einstein  of  the  probability  criterion  for 
numerical  coefficients,  it  is  nevertheless  to  be  regarded  as  small  con- 
sidering the  size  of  the  exponents  of  the  quantities  with  which  it  is 
associated,  |  and  it  is  undeniable  that  the  result  is  of  such  simplicity 
that  it  seems  probable  that  the  coefficient  may  be  the  result  of  a 
mathematical  process,  and  is  not  merely  due  to  a  chance  combina- 
tion of  elements  in  a  dimensionally  correct  form.  ! 

At  any  rate,  whatever  our  opinions  as  to  the  validity  of  the  argu- 
ment, the  striking  character  of  the  result  sticks  in  our  minds,  and 
we  reserve  judgment  until  the  final  solution  is  forthcoming,  in  the 
same  way  that  the  periodic  classification  of  the  elements  had  to  be 
carried  along  with  suspended  judgment  until  the  final  solution  was 
forthcoming.  It  may  be  mentioned  that  Lorentz  and  his  pupils  have 
tried  a  detailed  analysis  on  these  terms,  with  unsuccessful  results. 

The  above  analysis  gives  other  opportunities  for  thought.  It  is 
significant  that  the  quantum  h  does  not  enter  the  result,  although  it 
appears  to  be  inseparably  connected  with  the  radiation  processes, 
at  least  in  ponderable  matter.  We  know  that  h  enters  the  formula 
for  the  spectral  distribution  of  energy,  and  we  also  know  from 
thermodynamics  that  the  distribution  of  energy  throughout  the 
spectrum  in  a  cavity  such  as  the  above  is  the  same  as  the  distri- 
bution in  equilibrium  with  a  black  body  composed  of  atoms.  The 
spectral  distribution  in  the  cavity  which  we  have  been  considering 
must  therefore  involve  h.  Does  this  mean  that  h  can  be  determined 
in  terms  of  the  electronic  constants,  the  gas  constant,  and  the  con- 


96  DIMENSIONAL  ANALYSIS 

stants  of  the  ether,  so  that  no  mechanism  with  which  we  are  not 
already  familiar  is  needed  to  account  for  h?  Of  course  Lewis  has 
used  Planck's  formula  for  h  in  terms  of  a,  etc.,  in  order  to  obtain  a 
numerical  value  for  h  in  terms  of  other  quantities. 

As  an  additional  example  of  the  application  of  dimensional  analy- 
sis in  theoretical  investigations  let  us  examine  the  possibility  of 
explaining  the  mechanical  behavior  of  substances  on  the  basis  of  a 
particular  form  of  the  law  of  force  between  atoms.  "We  suppose  that 
the  law  of  force  can  be  written  in  the  form 

F  =  A  r~2  +  B  r~n. 

A  is  to  be  intrinsically  negative,  and  represents  an  attractive  force, 
and  B  is  positive,  and  represents  a  force  of  repulsion  which  becomes 
very  intense  on  close  approach  of  the  atoms.  The  atoms  of  different 
substances  may  differ  in  mass  and  in  the  numerical  value  of  the 
coefficients  A  and  B,  but  the  exponent  n  is  to  be  the  same  for  all 
substances.  We  also  suppose  that  the  temperature  is  so  high  that  the 
quantum  h  plays  no  important  part  in  the  distribution  of  energy 
among  the  various  degrees  of  freedom,  but  that  the  gas  constant 
is  sufficient  in  determining  the  distribution.  The  external  variables 
which  may  be  imposed  on  the  system  are  the  pressure  and  the  tem- 
perature. When  these  are  given  the  volume  is  also  determined,  and 
all  the  other  properties.  We  have,  therefore,  the  following  list  of 
quantities  in  terms  of  which  any  of  the  properties  of  the  substance 
are  to  be  determined. 

Name  of  Quantity.  Symbol.             Dimensional  Formula. 

Pressure,  p  ML-1T-2 

Temperature,  0  6 

Mass  of  the  atom,  m  M 

Gas  constant,  k  MLT"2^1 

A  (of  the  law  of  force),  A  ML3T~2 

B  (of  the  law  of  force),  B  MLn+1T-2 

In  addition  to  these  we  will  have  whatever  particular  property 
of  the  substance  is  under  discussion.  In  the  above  list  there  are  six 
quantities  in  terms  of  four  fundamental  units.  Therefore  from  this 
list  of  permanent  variables  there  are  two  dimensionless  products. 
Let  us  find  them.  We  will  choose  one  involving  p  and  not  0,  and  the 


APPLICATIONS  TO  THEORETICAL  PHYSICS         97 

other  6  and  not  p,  since  p  and  0  are  the  physical  variables  under 
our  control.  The  products  are  at  once  found  to  be 

_n  +  2  * 

pA  n-2  Bn-2 
and 

n  -1  1 

A~^2   B5TT2   k  6. 

The  existence  of  these  two  products  already  gives  us  information 
about  the  behavior  of  the  body  in  those  cases  in  which  pressure  and 
temperature  are  not  independently  variable  quantities,  as  they  are 
not  on  the  vapor  pressure  curve,  or  on  the  melting  curve,  or  on  the 
curve  of  equilibrium  between  two  allotropic  modifications  of  the 
solid.  Under  these  conditions  we  have 

n-f  2  4  /  n-1  1  \ 

p  A~^*  B^7*  =  f  lA"^^  Birr2  k  0/, 

where  f  is  the  same  function  for  all  substances.  A  and  B  vary  from 
substance  to  substance.  Hence  this  analysis  shows  that  in  terms  of 
a  new  variable  p  Cx  for  the  pressure,  and  a  new  variable  B  C2  for 
temperature,  the  equations  for  the  equilibrium  curves  of  all  sub- 
stances are  the  same.  These  new  pressure  and  temperature  variables 
are  obtained  by  multiplying  the  ordinary  pressure  and  temperature 
by  constant  factors,  and  may  be  called  the  reduced  pressure  and 
temperature.  Van  der  Waal's  equation  is  a  particular  case  of  such 
an  equation,  which  becomes  the  same  for  all  substances  in  terms  of 
the  reduced  variables. 

Now  consider  any  other  physical  property  of  the  substance  which 
is  to  be  accounted  for  in  terms  of  the  variables  of  the  analysis  above. 
We  have  to  form  another  dimensionless  product  in  which  it  is 
involved.  This  dimensionless  product  may  most  conveniently  be 
expressed  in  terms  of  the  quantities  m,  k,  A,  and  B,  since  these  are 
physically  invariable  for  the  particular  substance.  The  expression 
of  any  physical  quantity  is  always  dimensionally  possible  in  terms 
of  these  quantities,  unless  the  determinant  of  the  exponents  of  m,  k, 
A,  and  B  vanishes,  and  this  is  seen  to  vanish  only  in  the  case  n  = 
+2,  which  is  the  trivial  case  of  the  force  reducing  to  an  attraction 
alone.  Hence  in  the  general  case  any  physical  property,  which  we 
may  call  Q,  may  be  expressed  in  the  form 

(n+2  *  n-1  1  \ 

p  A~^~2  B^  A~^«  B5^  k  0), 


98  DIMENSIONAL  ANALYSIS 

where  the  Const  may  involve  m,  k,  A,  and  B  in  any  way,  but  does 
not  involve  p  or  6.  Now  if  we  define  Q/Const  as  the  ' ( reduced ' '  value 
of  Q,  then  we  have  the  important  result  that  for  all  substances  of 
this  type  the  equation  connecting  the  reduced  value  of  a  quantity 
Q  with  the  reduced  pressure  and  temperature  is  the  same.  This 
applies  not  only  to  thermodynamic  properties,  but  to  all  properties 
which  are  to  be  explained  in  terms  of  the  same  structure,  such  as 
thermal  conductivity  or  viscosity. 

The  values  of  the  factors  by  means  of  which  the  measured  values 
of  the  physical  variables  are  converted  to  "reduced"  values  will 
enable  us  to  compute  A,  B,  and  m  for  the  substance  in  question,  if  n 
can  be  otherwise  determined. 

It  is  evident  on  consideration  of  the  above  work  that  the  only 
assumption  which  we  have  made  about  n  is  that  it  is  dimensionless, 
and  that  we  have  not  used  the  assumption  stated  in  the  beginning 
that  n  is  the  same  for  all  substances.  We  may  therefore  drop  this 
assumption,  and  have  the  theorem  that  for  all  substances  whose 
behavior  can  be  determined  in  terms  of  atoms  which  are  character- 
ized by  a  mass  and  a  law  of  force  of  the  form  Ar~2  +  Br~n,  with  no 
restriction  on  A,  B,  or  n,  there  is  a  law  of  corresponding  states  for 
all  physical  properties.  \ 

Evidently  it  would  be  possible  to  carry  through  an  analysis  like 
the  above  in  which  the  external  variables  p  and  0  are  replaced  by 
any  other  two  which  might  be  convenient,  such  as  certain  of  the 
thermodynamic  potentials,  and  the  same  result  would  have  been 
obtained,  unless  there  should  happen  to  be  special  relations  between 
the  dimensional  exponents.  Whether  there  are  such  special  relations 
can  be  easily  determined  in  any  special  case. 

Before  anyone  starts  on  a  detailed  development  of  such  a  theory 
of  the  structure  of  matter  as  this,  he  would  make  a  preliminary 
examination  to  see  whether  the  properties  of  substances  do  actually 
obey  such  a  law  of  corresponding  states,  and  govern  his  future 
actions  accordingly.  The  value  of  the  advance  information  obtained 
in  this  way  is  incontestable. 

The  analysis  above  reminds  one  in  some  particulars  of  that  of 
Meslin,5  but  is  much  more  general,  in  that  the  analysis  of  Meslin 
applied  only  to  the  equation  of  state,  and  had  to  assume  the  exist- 
ence of  critical,  or  other  peculiar  points. 

As  a  final  application  of  dimensional  analysis  to  theoretical 


APPLICATIONS  TO  THEORETICAL  PHYSICS         99 

physics  we  consider  the  determination  of  the  so-called  absolute  sys- 
tems of  units. 

The  units  in  ordinary  use  are  ones  whose  absolute  size  is  fixed 
in  various  arbitrary  ways,  although  the  relations  between  the  differ- 
ent sorts  of  units  may  have  a  logical  ring.  Thus  the  unit  of  length, 
the  centimeter,  was  originally  denned  as  bearing  a  certain  relation 
to  a  quadrant  of  the  earth 's  circumference,  and  the  unit  of  mass  is 
the  mass  of  a  quantity  of  water  occupying  the  unit  volume.  There  is 
something  entirely  arbitrary  in  selecting  the  earth  and  water  as  the 
particular  substances  which  are  to  fix  the  size  of  the  units. 

We  have  also  met  in  the  course  of  our  many  examples  dimensional 
constants.  These  constants  usually  are  the  expression  of  some  pro- 
portionality factor  which  enters  into  the  expression  of  a  law  of 
nature  empirically  discovered.  Such  dimensional  constants  are  the 
constant  of  gravitation,  the  velocity  of  light,  the  quantum,  the  con- 
stant of  Stefan's  law,  etc.  Now  the  numerical  magnitude  of  the 
dimensional  constants  depends  on  the  size  of  the  fundamental  units 
in  a  way  fixed  by  the  dimensional  formulas.  By  varying  the  size  of 
the  fundamental  units,  we  may  vary  in  any  way  that  we  please  the 
numerical  magnitude  of  the  dimensional  constant.  In  particular,  by 
assigning  the  proper  magnitudes  to  the  fundamental  units  we  might 
make  the  numerical  magnitudes  of  certain  dimensional  constants 
equal  to  unity.  Now  the  dimensional  constants  are  usually  the 
expression  of  some  universal  law  of  nature.  If  the  fundamental  units 
are  so  chosen  in  size  that  the  dimensional  constants  have  the  value 
unity,  then  we  have  determined  the  size  of  the  units  by  reference  to 
universal  phenomena  instead  of  by  reference  to  such  restricted 
phenomena  as  the  density  of  water  at  atmospheric  pressure  at  some 
fixed  temperature,  for  instance,  and  the  units  to  that  extent  are 
more  significant. 

There  is  no  reason  why  one  should  be  restricted  to  dimensional 
constants  of  universal  occurrence  in  fixing  the  size  of  the  units,  but 
any  phenomenon  of  universal  occurrence  may  be  used.  Thus  the 
units  may  be  so  chosen  that  the  charge  on  the  electron  is  unity. 

Any  system  of  units  fixed  in  this  way  by  reference  to  phenomena 
or  relationships  of  universal  occurrence  and  significance  may  be 
called  an  absolute  system  of  units.  The  first  system  of  absolute  units 
was  given  by  Planck6  in  his  book  on  heat  radiation.  He  connected 
the  particular  system  which  he  gave  with  the  quantum,  and  it  might 
appear  from  Planck's  treatment  that  before  the  discovery  of  the 


100  DIMENSIONAL  ANALYSIS 

quantum  there  were  not  enough,  dimensional  constants  of  the  proper 
character  known  to  make  possible  a  universal  system  of  units,  but 
such  is  not  the  case.  Planck  was  the  first  to  think  of  the  possibility 
of  absolute  units,  and  used  the  quantum  in  determining  them,  but 
there  is  no  necessary  connection  with  the  quantum,  as  may  be  seen 
in  the  following  discussion. 

Let  us  now  determine  from  the  dimensional  formulas  the  set  of 
absolute  units  given  by  Planck.  To  fix  this  set  of  units  we  choose 
the  constant  of  gravitation,  the  velocity  of  light,  the  quantum,  and 
the  gas  constant.  "We  require  that  the  fundamental  units  be  of  such 
a  size  that  each  of  these  dimensional  constants  has  the  value  unity 
in  the  new  system.  The  discussion  may  be  simplified  for  the  present 
by  omitting  the  gas  constant,  for  this  is  the  only  one  which  involves 
the  unit  of  temperature,  and  it  is  obvious  that  after  the  units  of 
mass,  length,  and  time  have  been  fixed,  the  gas  constant  may  be 
made  unity  by  properly  choosing  the  size  of  the  degree.  In  deter- 
mining the  size  of  the  new  units  we  find  it  advantageous  to  choose 
the  form  of  notation  used  in  the  third  chapter  in  changing  units. 
Consider,  for  example,  the  constant  of  gravitation.  We  write  this  as 

Constant  of  gravitation  =  G  =  6.658  X  10~8  gmr1  cm3  see"2. 

The  value  in  the  new  system  of  units  is  to  be  found  by  substituting 
in  the  expression  for  G  the  value  of  the  new  units  in  terms  of  the 
old.  Thus  if  the  new  unit  of  mass  is  such  that  it  is  equal  to  x  gm, 
and  the  new  unit  of  length  is  equal  to  y  cm,  and  the  new  unit  of 
time  to  z  sec,  we  shall  have  as  the  equation  to  determine  x,  y,  and  z, 
since  the  numerical  value  of  the  gravitational  constant  is  to  be  unity 
in  the  new  system 

6.658  X  10~8  gm"1  cm3  see"2  =  1  (x  gm)-1  (y  cm)3  (z  sec)"2. 

The  other  two  dimensional  constants  give  the  two  additional  equa- 
tions needed  to  determine  x,  y,  and  z.  These  other  equations  are 
immediately  written  down  as  soon  as  the  dimensional  formulas  and 
the  numerical  values  of  the  velocity  of  light  and  the  quantum  are 
known.  The  equations  are 

3  X  1010  cm  sec"1  =  1  (y  cm)  (z  sec)"1 
6.55  X  10~27  gm  cm2  sec"1  =  1  (x  gm)  (y  cm)2  (z  sec)-1. 

This  set  of  three  equations  may  be  readily  solved,  and  gives  x  = 


APPLICATIONS  TO  THEORETICAL  PHYSICS       101 

5.43  X  10~5,  y  =  4.02  X  10~33,  and  z  =  1.34  X  10~43.  This  means 

that 

the  new  unit  of  mass  is  5.43  X  10~5  gm 
the  new  unit  of  length  4.02  X  10~33  cm 
the  new  unit  of  time      1.34  X  10~43  sec. 

So  far  all  is  plain  sailing,  and  there  can  be  no  question  with 
regard  to  what  has  been  done.  The  attempt  is  sometimes  made  to  go 
farther  and  see  some  absolute  significance  in  the  size  of  the  units 
thus  determined,  looking  on  them  as  in  some  way  characteristic  of  a 
mechanism  which  is  involved  in  the  constants  entering  the  defini- 
tion. Thus  Eddington7  says:  "There  are  three  fundamental  con- 
stants of  nature  which  stand  out  preeminently,  the  velocity  of  light, 
the  constant  of  gravitation,  and  the  quantum.  From  these  we  can 
construct  a  unit  of  length  whose  value  is  4  X  10~33  cm.  There  are 
other  natural  units  of  length,  the  radii  of  the  positive  and  negative 
charges,  but  these  are  of  an  altogether  higher  order  of  magnitude. 
"With  the  possible  exception  of  Osborne  Reynold's  theory  of  matter, 
no  theory  has  attempted  to  reach  such  fine  grainedness.  But  it  is 
evident  that  this  length  must  be  the  key  to  some  essential  structure. ' ' 

Speculations  such  as  these  arouse  no  sympathetic  vibration  in  the 
convert  to  my  somewhat  materialistic  exposition.  The  mere  fact  that 
the  dimensional  formulas  of  the  three  constants  used  was  such  as  to 
allow  a  determination  of  the  new  units  in  the  way  proposed  seems 
to  be  the  only  fact  of  significance  here,  and  this  cannot  be  of  much 
significance,  because  the  chances  are  that  any  combination  of  three 
dimensional  constants  chosen  at  random  would  allow  the  same  pro- 
cedure. Until  some  essential  connection  is  discovered  between  the 
mechanisms  which  are  accountable  for  the  gravitational  constant, 
the  velocity  of  light,  and  the  quantum,  it  would  seem  that  no  signifi- 
cance whatever  should  be  attached  to  the  particular  size  of  the  units 
defined  in  this  way,  beyond  the  fact  that  the  size  of  such  units  is 
determined  by  phenomena  of  universal  occurrence. 

Let  us  now  continue  with  our  deduction  of  the  absolute  units, 
and  introduce  the  gas  constant.  For  this  we  have  the  equation 

Gas  constant  =  k  =  2.06  X  10~16  gm  cm2  see"2  0"1 

1  (xgm)  (ysec)2  (zsec)~2  (wfl)-1. 

x,  y,  and  z  are  already  determined,  so  that  this  is  a  single  equa- 
tion to  determine  w.  The  value  found  is  2.37  X  1032.  This  means 


102  DIMENSIONAL  ANALYSIS 

that  the  new  degree  must  be  equal  to  2.37  X  1032  ordinary  Centi- 
grade degrees. 

In  the  wildest  speculations  of  the  astrophysicists  no  such  tem- 
perature has  ever  been  suggested,  yet  would  Professor  Eddington 
maintain  that  this  temperature  must  be  the  key  to  some  funda- 
mental cosmic  phenomenon? 

It  must  now  be  evident  that  it  is  possible  to  get  up  systems  of 
absolute  units  in  a  great  number  of  ways,  depending  on  the  univer- 
sal constants  or  phenomena  whose  numerical  values  it  is  desired  to 
simplify.  "With  any  particular  selection  of  constants,  the  method 
in  general  is  the  same  as  that  in  the  particular  case  above.  In  general 
there  will  be  four  fundamental  kinds  of  unit,  if  we  want  to  restrict 
ourselves  to  the  electrostatic  system  of  measuring  electrical  charges, 
and  define  the  magnitude  of  the  charge  in  such  a  way  that  the  force 
between  two  charges  is  equal  to  their  product  divided  by  the  square 
of  the  distance  between  them,  or  if  we  do  not  restrict  ourselves  to  the 
electrostatic  system,  there  may  be  five  fundamental  kinds  of  quan- 
tity. There  seems  to  be  nothing  essential  in  the  number  five,  which 
merely  arises  because  we  usually  find  it  convenient  to  use  the  me- 
chanical system  of  units  in  which  the  constant  of  proportionality 
between  force  and  the  product  of  mass  and  acceleration  is  always 
kept  fixed  at  unity.  The  convenience  of  this  system  is  perhaps  more 
obvious  in  the  case  of  mechanical  phenomena,  because  of  the  univer- 
sality of  their  occurrence.  But  if  temperature  effects  were  as  univer- 
sal and  as  familiar  to  us,  we  would  also  insist  that  we  always  deal 
only  with  that  system  of  units  in  which  the  gas  constant  has  the 
fixed  value  unity. 

Having,  therefore,  fixed  the  number  of  fundamental  units  which 
we  deem  convenient,  and  having  chosen  the  numerical  constants 
whose  values  we  wish  to  simplify,  we  proceed  as  above.  It  is  evident 
that  it  will  in  general  be  necessary  to  assign  as  many  constants  as 
there  are  fundamental  units,  for  otherwise  there  will  not  be  enough 
equations  to  give  the  unknowns.  Thus  above,  we  fixed  four  con- 
stants, gravitational,  velocity  of  light,  quantum,  and  gas  constant, 
and  we  had  four  fundamental  kinds  of  units.  Now  it  is  important 
to  notice  that  four  algebraic  equations  in  four  unknowns  do  not 
always  have  a  solution,  but  the  coefficients  must  satisfy  a  certain 
condition.  This  condition  is,  when  applied  to  the  dimensional  for- 
mulas into  which  the  unknowns  enter,  that  the  determinant  of  the 
exponents  must  not  vanish.  In  general,  a  four-rowed  determinant 


APPLICATIONS  TO  THEORETICAL  PHYSICS       103 

selected  at  random  would  not  be  expected  to  vanish.  In  the  case  of 
the  determinants  obtained  from  the  dimensional  formulas  of  the 
constants  of  nature  this  is  not  the  case,  however,  because  the  dimen- 
sional formulas  are  nearly  all  of  them  of  considerable  simplicity, 
and  the  exponents  are  nearly  always  small  integers.  It  very  often 
happens  that  the  determinant  of  the  exponents  of  four  constants 
chosen  at  random  vanishes,  and  the  proposed  scheme  for  determin- 
ing the  absolute  units  turns  out  to  be  impossible.  The  vanishing  of 
the  determinant  means  that  all  the  quantities  are  not  dimensionally 
independent,  so  that  we  really  have  not  four  but  a  smaller  number 
of  independent  quantities  in  terms  of  which  to  determine  the  un- 
knowns. For  instance,  we  have  found  that  the  gravitational  constant 
dimensionally  has  the  same  formula  as  the  square  of  the  ratio  of 
the  charge  to  the  mass  of  the  electron.  This  means  that  we  could 
not  set  up  a  system  of  absolute  units  in  which  the  gravitational 
constant,  the  charge  on  the  electron,  and  the  mass  of  the  electron 
were  all  equal  to  unity.  Now  let  us  write  down  some  of  the  impor- 
tant constants  of  nature  and  see  what  are  the  possibilities  in  the 
way  of  determining  systems  of  absolute  units. 

Gravitation  constant,  G  6.658  X  10"8  gm"1  cm3  sec~2 

Velocity  of  light,  c  3  X  1010  cm  sec"1 

Quantum,  h  6.547  X  10"27  gm-cm2  sec"1 

Gas  constant,  k  2.058  X  10"16  gm  cm2  see"2  °C-1 

Stefan  constant,  a  7.60  X  10~15  gm  cm"1  see"2  °C-4 

First  spectral  constant,  C  0.353  X  gm.  cm4  sec~3 

Second  spectral  constant,  a'  1.431  cm  °C 

Rydberg  constant,  R  3.290  X  1015  sec"1 

Charge  of  the  electron,  e  4.774  X  10"10  gm'  cm*  sec-1 

Mass  of  the  electron,  m  8.8  X  10~28  gm 

Avogadro  number,  N  6.06  X  1023  gm"1 

Second  Avogadro  number,  N'  7.29  X  1015  gm"1  cm~2  sec2  °C 

Some  of  the  quantities  in  the  above  list  require  comment.  The 
Stefan  constant  "a"  is  defined  by  the  relation  u  =  a04,  where  u  is 
the  energy  density  in  the  hohlraum  in  equilibrium  with  the  walls 
at  temperature  0.  The  first  and  second  spectral  constants  are  the 
constants  in  the  formula 


Cr*l  -1-1 

-vp-ij 


104  DIMENSIONAL  ANALYSIS 

for  the  distribution  of  energy  in  the  spectrum.  The  Avogadro  num- 
ber N  is  denned  as  the  number  of  molecules  per  gm  molecule,  and 
its  dimensions  may  be  obtained  from  the  formula  for  it ;  N  =  (no.  of 
molecules  per  gm)  X  (mass  of  molecule  /  mass  of  hydrogen  mole- 
cule). Its  dimensions  are  evidently  the  reciprocal  of  a  mass,  and  the 
numerical  value  is  merely  the  reciprocal  of  the  mass  of  the  hydro- 
gen molecule.  The  second  Avogadro  number  N'  is  denned  as  the 
number  of  molecules  per  cm3  in  a  perfect  gas  at  unit  temperature 
and  at  unit  pressure.  We  know  that  this  number  is  independent  of 
the  particular  gas,  and  is  therefore  suited  to  be  a  universal  constant. 
Its  dimensions  are  evidently  those  of  vol"1  pressure"1  temp,  and  the 
numerical  value  may  be  found  at  once  in  terms  of  the  other 
constants. 

We  have  now  a  list  of  twelve  dimensional  constants  in  terms  of 
which  to  define  an  absolute  system  of  units.  Since  these  constants 
are  defined  in  that  system  in  which  there  are  four  fundamental 
kinds  of  unit,  in  general  any  four  of  the  twelve  would  sufiice  for 
determining  the  absolute  system  of  units,  but  the  relations  are  so 
simple  that  there  are  a  large  number  of  cases  in  which  the  deter- 
minant of  the  exponents  vanishes,  and  the  choice  is  not  possible.  For 
instance,  C  has  dimensionally  the  same  formula  as  he2,  so  that  no 
set  of  four  into  which  C,  h,  and  c  all  enter  is  a  possible  set.  k  has  the 
dimensions  of  cha'"1,  so  that  the  set  k,  c,  h,  and  a'  is  not  possible. 
N'  has  the  dimensions  of  k"1,  so  that  no  set  of  four  into  which  both 
k  and  N'  enter  is  possible.  The  examples  might  be  continued  further. 
The  moral  is  that  it  is  not  safe  to  try  for  a  set  of  absolute  units  in 
terms  of  any  particular  group  of  constants  until  one  is  assured  that 
the  choice  is  possible.  For  instance,  one  set  that  might  seem  quite 
fundamental  turns  out  to  be  impossible.  It  is  not  possible  so  to 
choose  the  magnitudes  of  the  units  that  the  velocity  of  light,  the 
quantum,  the  charge  on  the  electron,  and  the  gas  constant  all  have 
the  value  unity. 

By  way  of  contrast,  certain  sets  which  are  possible  may  be  men- 
tioned. It  will  be  found  that  the  determinant  of  the  exponents  of 
the  following  does  not  vanish;  G,  c,  h,  k;  G,  c,  e,  k;  N,  c,  h,  k; 
N,  c,  e,  k. 

If  one  has  certain  criteria  of  taste  which  make  certain  of  the  above 
list  of  quantities  objectionable  as  universal  constants,  somewhat 
startling  results  may  be  obtained.  Let  us  decline  to  consider  the 
quantities  R,  m,  N,  and  N'.  There  remain  G,  c,  h,  k,  a,  C,  a',  and  e. 


APPLICATIONS  TO  THEORETICAL  PHYSICS       105 

Now  it  will  be  found  that  the  last  seven  of  these  have  the  property 
that  it  is  not  possible  to  choose  any  four  of  them  whose  exponential 
determinant  does  not  vanish.  Hence  any  set  of  four  quantities  in 
terms  of  which  the  absolute  system  of  units  is  to  be  determined,  if 
selected  from  the  above  list  of  eight,  must  include  the  gravitational 
constant.  This  fact  is  what  has  made  possible  Tolman's  Principle  of 
Similitude.8  It  seems  to  me  that  it  is  not  possible  to  ascribe  any 
significance  to  the  fact  that  there  exist  these  relations  between  the 
various  dimensional  constants,  but  it  must  be  regarded  as  an  entirely 
fortuitous  result  due  to  the  limited  number  of  elements  of  which  the 
dimensional  formulas  are  composed,  and  their  relative  simplicity. 

Another  interesting  speculation  on  the  nature  of  the  absolute 
units  requires  comment.  G.  N.  Lewis4  has  stated  it  to  be  his  convic- 
tion that  any  set  of  absolute  units  will  be  found  to  bear  a  simple 
numerical  relation  to  any  other  possible  set  of  absolute  units.  The 
justification  of  this  point  of  view  at  present  is  not  to  be  found  in 
any  accurate  results  of  measurement,  but  is  rather  quasi-mystical 
in  its  character.  This  point  of  view  led  Lewis  to  notice  the  remark- 
ably simple  relation  between  the  Stefan  constant  and  the  electronic 
charge  and  the  gas  constant,  but  so  far  as  I  know  it  has  not  been 
fruitful  in  other  directions,  and  I  have  already  indicated  another 
possible  significance  of  the  simplicity  of  the  relation. 

Now  let  us  examine  this  hypothesis~of  Lewis's  with  a  numerical 
example.  We  have  already  found  the  magnitude  of  the  fundamental 
units  which  would  give  the  value  unity  to  the  gravitational  con- 
stant, the  velocity  of  light,  the  quantum,  and  the  gas  constant.  Let 
us  now  find  what  size  units  would  make  the  gravitational  constant, 
the  velocity  of  light,  the  gas  constant,  and  the  charge  on  the  electron 
all  equal  to  unity.  The  work  is  exactly  the  same  in  detail  as  before, 
and  it  is  not  necessary  to  write  out  the  equations  again.  It  will  be 
found  that  the  following  units  are  required. 

New  unit  of  mass,  1.849  X  10~6  gm 

New  unit  of  length,  1.368  X  10~34  cm 

New  unit  of  time,  4.56  X  10~45  sec 

New  unit  of  temperature,  8.07  X  1030  °C 

Now  the  ratio  of  all  these  units  to  the  ones  previously  determined 
will  be  found  to  be  1/29.36.  On  the  face  of  it,  29.36  does  not  appear 
to  be  a  particularly  simple  number,  but  on  examining  the  way  in 
which  it  came  into  the  formulas,  it  will  be  found  that  29.36  is  the 


106  DIMENSIONAL  ANALYSIS 

approximate  value  of  4  TT  (  - — )  ,  and  this  somewhat  complicated 

\  15  / 

numerical  expression  came  from  Planck 's  relation  between  Stefan 's 
constant  and  the  spectral  radiation  constants.  In  fact,  using  Lewis's 
value  for  a,  Planck's  formula  for  h  becomes 


It  would  seem  that  there  will  be  considerable  hesitation  in  calling 
a  numerical  coefficient  of  this  form  "simple."  If  this  is  simple,  it 
is  hard  to  see  what  the  criterion  of  numerical  simplicity  is,  and 
Lewis's  principle,  at  least  as  a  heuristic  principle,  becomes  of  ex- 
ceedingly doubtful  value.  Lewis 's9  own  feeling  is  that  the  coefficient 
in  the  above  form  cannot  be  regarded  as  simple,  and  the  fact  that  it 
cannot  is  presumptive  evidence  that  the  formula  as  given  by  Planck 
can  be  regarded  only  as  an  approximation,  and  that  sometime  a 
more  rigorous  theory  will  be  possible  in  which  the  number  which  is 
at  present  within  the  experimental  error  equal  to  29.36  will  be 
expressed  in  a  way  which  will  appeal  to  everyone  as  simple  as  made 
up  of  simple  integers  and  TT'S. 

The  justification  of  such  speculations  is  thus  for  the  future.  The 
spirit  of  such  speculations  is  evidently  opposed  to  the  spirit  of  this 
exposition,  and  we  are  for  the  present  secure  in  our  point  of  view 
which  sees  nothing  mystical  or  esoteric  in  dimensional  analysis. 

BEFERENCES 

(1)  A.  Einstein,  Ann.  Phys.  35,  686,  1911. 

(2)  J.  H.  Jeans,  Trans.  Roy.  Soc.  201  (A),  157,  1903. 

(3)  J.  H.  Jeans,  Proc.  Roy.  Soc.  76,  545,  1905. 

(4)  G.  N.  Lewis  and  E.  Q.  Adams,  Phys.  Rev.  3,  92,  1914. 

(5)  Meslin,  C.  R.  116,  135,  1893. 

(6)  M.  Planck,  Heat  Radiation,  English  translation  by  Masius, 
p.  175. 

(7)  A.  S.  Eddington,  Report  on  Gravitation,  Lon.  Phys.  Soc., 
1918,  p.  91. 

(8)  R.  C.  Tolman,  Phys.  Rev.  3,  244,  1914. 

(9)  Private  communication  from  Lewis. 


PROBLEMS 

1.  THE  gas  constant  in  the  equation  pv  =  RT,  has  the  value  0.08207 
when  the  pressure  p  is  expressed  in  atmospheres,  the  volume  v  is  the 
volume  in  liters  of  1  gm  mol,  and  T  is  absolute  Centigrade  degrees. 
What  is  R  when  p  is  expressed  in  dynes/cm2  and  v  is  in  cm3  ? 

2.  The  thermal  conductivity  of  copper  is  0.92  cal.  per  cm2  per  sec 
per  1°  per  cm  temperature  gradient.  What  is  it  in  B.T.U.  per  hour 
per  square  foot  for  a  temperature  gradient  of  1°  Fahrenheit  per 
foot?  (This  last  is  the  engineering  unit.) 

3.  If  the  numerical  value  of  e2/ch  is  0.001161  in  terms  of  the  gm, 
cm,  and  sec,  what  is  its  value  in  terms  of  the  ton,  mile,  and  hour? 
e  is  the  charge  of  the  electron  in  E.S.U.,  c  is  the  velocity  of  light  in 
empty  space,  and  h  is  Planck's  quantum  of  action. 

4.  The  thrust  exerted  by  an  air  propeller  varies  with  the  number 
of  revolutions  per  second  and  the  speed  of  advance  along  the  axis 
of  revolution.  Show  that  the  critical  speed  of  advance  at  which  the 
thrust  vanishes  is  proportional  to  the  number  of  revolutions  per 
second. 

5.  Show  that  the  acceleration  toward  the  center  of  a  particle 
moving  uniformly  in  a  circle  of  radius  r  is  Const  v2/r. 

6.  Show  that  the  time  of  transverse  vibration  of  a  heavy  stretched 
wire  is  Const  X  length  X  (linear  density  /tension  )*. 

7.  The  time  of  longitudinal  vibration  of  a  bar  is  Const  X  length 
X  (volume  density /elastic  constant)*. 

8.  The  velocity  of  sound  in  a  liquid  is  Const  X  (density/modulus 
of  compressibility)'. 

9.  Given  that  the  twist  per  unit  length  of  a  cylinder  varies  in- 
versely as  the  elastic  constant,  or  as  the  moment  of  the  applied 
force,  prove  that  it  also  varies  inversely  as  the  fourth  power  of  the 
diameter. 

10.  There  is  a  certain  critical  speed  of  rotation  at  which  a  mass 
of  incompressible  gravitating  fluid  becomes  unstable.  Prove  that 
the  angular  velocity  at  instability  is  independent  of  the  diameter 
and  proportional  to  the  square  root  of  the  density. 

11.  There  is  a  certain  size  at  which  a  solid  non-rotating  gravitat- 
ing sphere  becomes  unstable  under  its  own  gravitation.  Prove  that 
the  radius  of  instability  varies  directly  as  the  square  root  of  the 
elastic  constant  and  inversely  as  the  density. 

12.  Given  that  the  velocity  of  advance  of  waves  in  shallow  water 
is  independent  of  the  wave  length,  show  that  it  varies  directly  as  the 
square  root  of  the  depth. 

13.  The  velocity  of  capillary  waves  varies  directly  as  the  square 


108  DIMENSIONAL  ANALYSIS 

root  of  the  surface  tension,  and  inversely  as  the  square  root  of  the 
wave  length  and  the  density. 

14.  A  mass  attached  to  a  massless  spring  experiences  a  damping 
force  proportional  to  its  velocity.  The  mass  is  subjected  to  a  periodic 
force.  Show  that  the  amplitude  of  vibration  in  the  steady  state  is 
proportional  to  the  force. 

15.  The  time  of  contact  of  two  equal  spheres  on  impact  is  propor- 
tional to  their  radius.  Given  further  that  the  time  varies  inversely 
as  the  fifth  root  of  the  relative  velocity  of  approach,  show  that  it 
varies  as  the  2/5th  power  of  the  density,  and  inversely  as  the  2/5th 
power  of  the  elastic  constant. 

16.  The  specific  heat  of  a  perfect  gas  (whose  atoms  are  character- 
ized by  their  mass  only)  is  independent  of  pressure  and  tempera- 
ture. 

17.  Show  that  if  a  gas  is  considered  as  an  assemblage  of  molecules 
of  finite  size  exerting  no  mutual  forces  on  each  other  except  when  in 
collision  the  viscosity  is  independent  of  the  pressure  and  is  propor- 
tional to  the  square  root  of  the  absolute  temperature. 

18.  Show  that  if  the  thermal  conductivity  of  the  gas  of  problem 
17  is  independent  of  the  pressure  it  is  also  proportional  to  the 
square  root  of  the  absolute  temperature. 

19.  A  periodic  change  of  temperature  is  impressed  on  one  face  of 
a  half-infinite  solid.  Show  that  the  velocity  of  propagation  of  the 
disturbance  into  the  solid  is  directly  as  the  square  root  of  the  fre- 
quency, and  the  wave  length  is  inversely  as  the  square  root  of  the 
frequency.  The  disturbance  sinks  to  1/eth  of  its  initial  value  in  a 
number  of  wave  lengths  which  is  independent  of  the  frequency  and 
the  thermal  constants  of  the  material. 

20.  A  long  thin  wire  is  immersed  in  a  medium  by  which  its  ex- 
ternal surface  is  maintained  at  a  constant  temperature.  Heat  is 
supplied  to  the  wire  by  an  alternating  current  of  telephonic  fre- 
quency at  the  rate  Q  coswt  per  unit  volume.  Show  that  the  ampli- 
tude of  the  periodic  fluctuation  of  the  average  temperature  of  the 
wire  is  of  the  form  0  r=  Q  d2/k  f  (w  c  d2/k),  where  d  is  the  diameter 
of  the  wire,  k  the  thermal  conductivity,  and  c  the  heat  capacity  per 
unit  volume.  If  the  wire  is  thin,  show  by  a  consideration  of  the 
numerical  values  of  k  and  c  for  metals  that  0  is  independent  of  o> 
and  c  and  assumes  the  approximate  form  6  =  Const  Q  d2/k. 

21.  The  internal  energy  of  a  fixed  quantity  of  a  perfect  gas, 
reckoned  from  0°  Abs.  and  0  pressure,  is  independent  of  the  pres- 
sure and  proportional  to  the  absolute  temperature.  Hence  the  inter- 
nal energy  reckoned  from  an  arbitrary  temperature  and  pressure 
as  the  initial  point  is  independent  of  pressure  and  proportional  to 
the  excess  of  the  absolute  temperature  over  that  of  the  initial  point. 

22.  Why  may  not  the  argument  of  problem  21  be  applied  to  the 
entropy  of  a  fixed  amount  of  a  perfect  gas  ? 

23.  T.  W.  Richards,  Jour.  Amer.   Chem.   Soc.   37,   1915,  finds 
empirically  the  following  relation  for  different  chemical  elements 


PROBLEMS  109 

ft  =  0.00021  A/D1-26  (Tm  -  50°) 
where  8  =  -(—]  is  the  compressibility,  A  is  the  atomic  weight,  D 


is  the  density,  Tm  the  melting  temperature  on  the  absolute  Centi- 
grade scale.  What  is  the  minimum  number  of  dimensional  constants 
required  to  make  this  a  complete  equation,  and  what  are  their 
dimensions  ? 

24.  Show  that  the  strength  of  the  magnetic  field  about  a  magnetic 
doublet  varies  inversely  as  the  cube  of  the  distance,  and  directly  as 
the  moment  of  the  doublet. 

25.  What  are  the  dimensions  of  the  dielectric  constant  of  empty 
space  in  the  electromagnetic  system  of  units  ?  What  is  its  numerical 
value  ? 

26.  What  are  the  dimensions  of  the  magnetic  permeability  of 
empty  space  in  the  electrostatic  system  of  units  ?  What  is  its  numeri- 
cal value? 

27.  Given  a  half  -infinite  conducting  medium  in  the  plane  surface 
of  which  an  alternating  current  sheet  is  induced.  Show  that  the 
velocity  of  propagation  of  the  disturbance  into  the  medium  varies 
as  the  square  root  of  the  specific  resistance  divided  by  the  periodic 
time,  and  the  extinction  distance  varies  as  the  square  root  of  the 
product  of  specific  resistance  and  the  periodic  time. 

28.  Show  that  the  self-induction  of  a  linear  circuit  is  proportional 
to  the  linear  dimensions. 

29.  A  sinusoidal  E.M.F.  is  applied  to  one  end  of  an  electrical  line 
with  distributed  resistance,  capacity,  and  inductance.  Show  that  the 
velocity  of  propagation  of  the  disturbance  is  inversely  proportional, 
and  the  attenuation  constant  is  directly  proportional  to  the  square 
root  of  the  capacity  per  unit  length. 

30.  An  electron  is  projected  with  velocity  v  through  a  magnetic 
field  at  right  angles  to  its  velocity.  Given  that  the  radius  of  curva- 
ture of  its  path  is  directly  proportional  to  its  velocity,  show  that  the 
radius  of  curvature  is  also  proportional  to  the  mass  of  the  electron, 
and  inversely  proportional  to  the  field  and  the  charge. 

31.  In  all  electrodynamical  problems  into  whose  solution  the 
velocity  of  light  enters,  the  unit  of  time  may  be  so  defined  that  the 
velocity  of  light  is  unity,  and  two  fundamental  units,  of  mass  and 
time,  suffice.  Write  the  dimensions  of  the  various  electric  and  mag- 
netic quantities  in  terms  of  these  units.  Obtain  the  formula  for  the 
mass  of  an  electron  in  terms  of  its  mass  and  radius.  .   .   .  Problems 
involving  the  gravitational  constant  may  also  be  solved  with  only 
the  units  of  mass  and  length  as  fundamental.  Discuss  the  formula 
for  the  mass  of  the  electron  with  gravitational  units. 

32.  The  Rydberg  constant  (of  the  dimensions  of  a  frequency)  de- 
rived by  Bohr  's  argument  for  a  hydrogen  atom  is  of  the  form  N  = 
Const  m  e*/h3,  where  e  and  m  are  the  mass  and  the  charge  of  the 
electron,  and  h  is  Planck's  quantum  of  action. 


INDEX 


Absolute  significance  of  relative  mag- 
nitude, 20,  21. 
Absolute  units,  99-106. 
Adams,  95,  106. 
Airplane,  84,  85. 
Angle,  dimensions  of,  3. 
Atomic  forces,  4,  96. 
Avogadro's  number,  103. 

Boussinesq,  9. 

Buchholz,  35. 

Buckingham,  24,  40,  46,  55,  81,  87. 

Bureau  of  Standards,  81,  87. 

Change  ratio,  29. 
Changing  units,  28-35. 
Complete  equation,  37. 
Corresponding  states,  97-98. 
Cowley,  87. 
Crehore,  80. 

Debye,  90. 

Determinant  of  exponents,  34,  43,  59, 

68,  103. 

Dielectric  constant,  78. 
Dimension,  23. 
Dimensional  constant,  14,  15,  36,  52, 

64. 

Dimensional  formulas,  23,  24. 
Dimensional  homogeneity,  41. 
Dimensionless  products,  40,  43. 
Drop  of  liquid,  3. 

Eddington,  101,  106. 
Edwards,  87. 
Einstein,  89,  95,  106. 
Elastic  pendulum,  59-65. 
Electromagnetic  mass,  12,  53. 
Electronic  charge,  103. 
Electronic  mass,  103. 
Energy  density,  77-79. 


Equations  of  motion,  52. 
Euler's  theorem,  39. 

Falling  sphere,  65-67,  84. 

Faraday,  77. 

Fessenden,  27,  80. 

Fitzgerald,  80. 

Fourier,  51,  55. 

Fractional  exponents,  46,  47. 

Gas  constant,  71,  72,  100,  101,  103. 
Gravitating  bodies,  5. 
Gravitation  constant,  6,  90,  100,  103. 
Gravitational  instability,  90. 
Gravity  wave,  56-59. 

Heat  transfer,  9,  72. 
Hersey,  87. 

Infra  red  frequency,  89. 
Instruments,  87. 

Jeans,  40,  46,  90,  93,  106. 
Johnson,  80. 

Levy,  87. 

Lewis,  95,  96,  105,  106. 

Lodge,  A.,  35. 

Lodge,  O.,  80. 

Logarithmic  constant,  36,  73-75. 

Lorentz,  95. 

Meslin,  98,  106. 
Millikan,  67. 
Models,  82,  85,  86. 

National  Physical  Laboratory,  81. 

Nernst,  74. 

Number  of  fundamental  units,  11,  24, 

48,  54,  55,  61,  63,  66,  68. 
Numerical  coefficients,  88-90. 


112 


INDEX 


Pendulum,  1,  81. 
Physical  relations,  91,  92. 
,Pi  theorem,  36-47,  40,  49,  57,  75. 
Planck,  96,  99,  100,  106. 
Pressure  of  gas,  70-71. 
Primary  quantities,  18-20,  23. 
Principle    of    Similitude,    Eayleigh's, 

10. 

Principle  of  Similitude,  Tolman  's,  105. 
Product  of  powers,  8,  22,  39. 
Projectile,  86. 

Quantum,  91,  95,  99,  100,  103. 

Eadiation  from  black  body,  93. 
Eayleigh,  4,  9,  10,  11,  24,  56,  69,  72, 

80,  87. 

Eeduced  variable,  97,  98. 
Eesistance  to  submerged  motion,  82- 

86. 

Eeynolds,  87,  101. 
Eiabouchinsky,  10,  11,  24. 
Eichardson,  93. 
Eouth,  46. 
Eiicker,  26,  80. 
Eydberg  constant,  103. 


Scattering  of  sky  light,  69. 
Secondary  quantities,  19-20,  22,  23. 
Spectral  constants,  103. 
Stefan,  95,  103,  105,  106. 
Stiffness  of  beam,  67-69. 
Stokes,  65,  84. 

Temperature,  dimensions  of,  10-11,  24, 

71,  72,  73. 

Thermal  conductivity,  92. 
Thomson,  James,  29. 
Thompson,  S.  P.,  27,  46. 
Time  constant  of  electric  circuit,  76. 
Tolman,  24,  26,  27,  105,  106. 
Transcendental  functions,  42,  44,  45. 
"True"  dimensions,  24,  79. 

Ultimate  rational  units,  105. 
Uniqueness  of  dimensions,  25,  26. 

Velocity  of  light,  12,  53,  100,  103. 

Webster,  35. 
Williams,  26,  46,  80. 
Wilson,  87. 


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